biological systems display fascinating spatial and temporal patterns , which hint that the underlying cellular processes are highly dynamic and operate on a wide range of time and length - scales .this is indeed confirmed by measurements of the temporal dynamics of protein concentrations and gene expression levels for various signalling and response systems , made using pulse - labelling , -galactosidase measurements and immunoblotting , electromobility shift assays , real time pcr , fluorescence techniques , chromatin immunoprecipitation assays and microarrays .overall , it is now evident that regulatory and signal transduction networks do not depend merely on shifting the relevant protein concentrations from one steady state level to another .rather , the signals often have a significant temporal variation that carries much more information and propogates through the regulatory networks in a complex manner . with time - resolved data now available for a number of response and signalling systems , it is perhaps the appropriate time to explore whether there are any commonalities , or design principles " , in the underlying mechanisms . in this reviewwe will show that a prominent subclass of such systems does indeed have a common underlying design structure which combines a negative feedback loop with a time delay .this subclass consists of systems that display oscillatory behaviour under some conditions .the most obvious examples are circadian rhythms and cell division .oscillations are also seen in the levels of cellular calcium . and in embryo development .somite segmentation , for instance , exhibits clearly periodic spatial patterns which are produced by periodic temporal variation of proteins like hes1 , axin , notch and wnt .we also include in this class , systems which display damped oscillations or semi - periodic behaviour .one example is oscillations , triggered by dna damage , in p53 , a key protein involved in cell death and apoptosis pathways . hormones , such as the human growth factor , also show such intermittently periodic behaviour and pulsatile secretion .for these systems the recurrent behaviour probably has a direct physiological role . in cyanobacteria , for example , various physiological processes , like respiration and carbohydrate synthesis , are directly influenced by the circadian clock to be in sync with the day - night cycle .for other systems , however , clear oscillatory behaviour is not observed in the wild - type , but only in certain mutants .for instance , the nf- signalling system , involved in immune response in mammalian cells , shows oscillations in nuclear nf- concentration only in a mutant which contains just one isoform of the nf- inhibitor , i , and not the other isoforms .fluorescence measurements of the nf- system modified so that i was overexpressed also showed sustained oscillations over several hours .however , wild - type cells show at best damped oscillations . hereit is not clear if the oscillations themselves have a significant physiological consequence or are merely a by - product of other requirements for the wild - type behaviour . nevertheless , the sub - parts of these systems which are potentially oscillatory are important , often essential , components which influence the complex temporally varying wild - type response .b , data taken from ref ..,width=302 ]  we will focus on three of the regulatory systems mentioned above : hes1 in mammalian embryos , p53-mdm2 in mammalian cells , and the nf- signalling system , also in mammalian cells .[ fig_oscill ] shows the oscillations observed in experiments for all three .these systems are quite complicated , with many components interacting in various ways including transcriptional activation / repression , translation and post - translation regulation , protein - protein interaction , targeted protein degradation , and active nuclear - cytoplasmic translocation composed into a complex network with multiple feedback loops . in this review, we mainly describe the mechanisms and structures in these networks that allow them to produce the observed oscillations .we will do this using simplified mathematical models where the level of description balances the need to correctly describe the systems with the need to coarse - grain over some details in order to reveal common design features .we first begin by defining the overall framework of models we consider , and describing the basic ingredients for producing oscillatory behaviour within this framework _ negative feedback _ and _ time delays_. in the subsequent sections we show how the three examples of he s , p53 and nf- contain these ingredients .therein we also elaborate , using stability analyses , the requirements for producing oscillations in these systems .finally , we briefly discuss how to extract information about underlying structures from oscillatory time series data .fig . [ schematic]a shows a schematic negative feedback loop .the individual nodes in this loop are the relevant dynamical variables : they could be protein concentrations , gene expression levels , etc .each variable either activates or represses the next one in the loop . by an activator , we mean that if the concentration of protein 1 goes up it tends to increase the concentration of protein 2 ( either by increasing its production rate or decreasing its degradation rate ) .a repressor has the opposite effect .then , a negative feedback loop is simply defined as _ a loop with an odd number of repressors _ , so that the effect of a perturbation in the concentration of any species in the loop eventually feeds back to itself with a negative sign .feedback loops are , in general , the most common network motifs in cellular organization , especially when one considers the regulation of small molecules .we concentrate on negative feedback here because it was hypothesized by thomas , and rigorously proven for a wide class of systems , that the presence of at least one negative feedback loop is a necessary condition for oscillations .the simplest negative feedback loop is of course a protein which represses itself .there are many examples of such proteins : the main repressor of the sos regulon in _e. coli _ , lexa , also represses its own production ; hes1 , mentioned above , also represses transcription of its own gene , shown schematically in fig .[ schematic]b .however , hes1 looks like a one - component loop only if we coarse - grain over the intermediate steps in protein production .if we count the hes1 mrna also then it becomes the two component negative feedback loop of fig .[ schematic]c : one component , he s protein , represses the production of the second , hes1 mrna , while the second component activates the first .the other two systems we discuss later in this review have the same loop structure when coarse - grained to the two component level .nuclear nf- is known to activate production of i , which inhibits nuclear import of nf- by sequestering it in the cytoplasm .p53 and mdm2 work similarly , with p53 activating the _ mdm2 _ gene and mdm2 sequestering p53 .  to examine the possible dynamical behaviour produced by a negative feedback loop , we model the dynamics of the concentrations of the components using ordinary coupled differential equations .for instance , for the two component loops shown in fig .[ schematic]c : where and are the concentrations of the two components and .this can easily be generalized to longer loops with components : which models a single feedback loop with no cross - links because the rate of change of a given variable depends only on itself and the preceding variable , . in writing such an equationwe are assuming that fluctuations in space and time are negligible .thus , there are no stochastic or diffusion terms. the functions model both production and degradation of the components , and can take many forms depending on the kind of interactions in the system .for example , in the p53 example ( with p53 , mdm2 ) , we know that p53 binds to an operator site at the promoter for the _ mdm2 _ gene and aids transcription. then , under some assumptions the production of mdm2 can be modeled using a term of the form : a sigmoidal monotonically increasing function of the p53 concentration , ( the hill coefficient , , accounts for cooperativity in the binding of p53 to the operator ) .it is difficult to say anything about the behaviour of the general coupled differential equation of a negative feedback loop , like eq .[ generaln ] , however , it is reasonable to constrain the functions of eq .[ generaln ] to be monotonic in .this corresponds to saying that a protein that activates a particular process can not change to repress it at some other concentration , and vice versa , which is the case for most transcription factors .promoter at low concentrations , but represses it at high concentrations .another example is the galactose regulator galr , which at high concentrations of galactose activates the promoter _galp2 _ but in the absence of galactose forms a dna loop , which completely represses _ galp2 _ .such examples are , however , somewhat rare and we will not consider them .] for monotone systems , not only is there no ambiguity about whether the loop implements negative or positive feedback , but we can also prove rigorously that there is only one fixed point ( see appendix a ) .the question then is whether such a steady state is stable or unstable . for two variables ,if the fixed point is unstable , and the system s trajectories are bounded ( again , a reasonable assumption for a biological system ) then it must show periodic oscillations : this is known as the poincar - bendixson theorem . for monotone systemsthis is true even when the loop has more than two components , which means that the question of whether oscillations are possible boils down to whether the fixed point is stable or unstable .physically , what is required for instability of the fixed point , and hence oscillations , is a time delay , or a slowing down of the signal going round the loop . by signal we mean perturbations of the concentrations away from the steady state .if a perturbation in the concentration of one variable instantaneously affects the concentration of the next one , and so on , then for a negative feedback loop , any perturbation will be immediately cancelled and the steady state will be stable . a sufficiently large time delay on the other hand will produce oscillations .this can be readily understood with an example : consider a person who walks along a straight line to a given point , marked on the ground .if this person is able to take instantaneous decisions , he will approach the mark and then stop .this is a stationary solution to the walk kinetics .if , on the other hand , it takes some time to realise that the mark has been reached , the person will not stop at the mark , but cross it .when eventually the information that the mark has been crossed is processed , the person will turn back and walk in the opposite direction .the mark will again be reached and overshot , and so on .the resulting kinetics will be damped or sustained oscillations about the mark . in cellular systemsmany processes could produce time delays .table 1 lists some timescales associated with such processes .  with the above insight, one can see that a simple way to model oscillations using the framework of eq .[ general ] is to introduce an explicit time delay into the equations : of course , this is the most general form , and often it is possible to have fewer than four delays .oscillations are observed even in the simplest case of a linear differential equation with a single ( sufficiently large ) delay ( see appendix c ) : , which models the 1-component he s loop of fig . [ schematic]b where hes1 represses itself after a time delay . although a linear delay rate equation is an oversimplification of hes1 production , the physics which lies behind more general delay differential equations like eq .( [ general_delay ] ) is the same .the added complication is that the functions and are in general highly nonlinear , resulting in an amplification of the effect of the delay .  putting an explicit time delay like this , of course , does not really shed light on the mechanism producing the time delay .there are several possibilities which can be used to produce oscillations in a negative feedback loop : +  1 .  a process that takes a finite minimum time 2 .many intermediate steps 3 .  a sharp response by some of the variables 4 .saturated degradation 5 .autocatalysis +  to elaborate : +  1 .rate equations , like eq .[ generaln ] , typically model processes which occur with a given average rate , such as the binding of a protein to an operator site .a hidden assumption is that the time interval between two binding events is poisson distributed , which means that often there is a reasonable probability for two events to be separated by a very short time interval ( say , much shorter than the average time between events ) .sometimes , however , molecular processes take a certain minimum time . for instance , if transcription and translation take a time after a polymerase binds to the promoter , then the rate of production of the protein is more appropriately modeled as , where is 1 if the polymerase is bound to the promoter at time , and zero otherwise .such logic has been used to justify time - delay models in a variety of systems .this is the approach we will take to model the p53 and he s systems , discussed in the subsequent sections .processes like transcription and translation have this character because they are in fact composed of a large number of intermediate processes : the polymerase binds , first forming a closed complex , which then makes a transition to an open complex , and then to an elongating complex , followed by many steps " along the dna until the polymerase reaches the end of the gene .even if each of these individual steps is a poisson process , the net effect adds up to a time delay .thus , instead of putting in an explicit time delay as in eq .[ general_delay ] , one could work with a negative feedback loop with many components .one simple example of such an oscillator is the repressilator , which is a negative feedback loop with six components .the repressilator also has another necessary ingredient a nonlinearity in the functions which allows some of the variables to respond to changes in the preceding variable in a faster - than - linear fashion .more precisely , the repressilator uses sigmoidal functions , like the function [ mm ] , to describe transcriptional repression , and needs hill coefficients of at least 2 in order to achieve oscillations .the earliest example of a negative feedback oscillator which uses a nonlinearity like this to produce a sharp response is a model by goodwin where the hill coefficient needs to be more than 8 .such high hill coefficients are unlikely for biological systems , however . in order to get around this problem " , bliss , painter andmarr introduced another way of producing an effective time delay .they used the saturated degradation of one of the concentrations .saturated degradation means that there is an upper limit to the degradation rate of one species , thereby allowing it to remain abundant for a longer time , thus effectively slowing down the signal travelling around the loop .such saturated degradation is quite common in biological systems , especially when proteins are tagged for targeted degradation by another protein , as we will show in the nf- case discussed below .finally , autocatalysis , where a molecule activates its own production can be used to produce oscillations in systems like eq .[ general ] .in fact for two variable systems where there is no explicit time - delay this is a necessary condition for oscillations .note that this modification typically makes the system non - monotonic .this survey of the theoretical requirements for producing oscillations in negative feedback loops already allows us to make an interesting observation : monotone 2-component loops without an explicit time - delay can not oscillate , whatever the nonlinearity in the functions .we prove this explicitly in appendix a. thus , if one insists on modelling an oscillating system using two variables , one must choose between introducing a time delay , and sacrificing monotonicity ..time scales of some cellular processes associated with a single molecule .the upper part of the table indicate the processes which are usually neglected in writing the rate equations of regulation newtworks , while the lower part indicate proceeses which are usually accounted for .[ cols= " < , < " , ]b ( c).,width=377 ]  we now turn to three biological systems to illustrate these ideas in action . in the following , we will briefly give a description of the systems , p53-mdm2 , hes1 and nf- , in which ultradian " oscillations have been observed , which have time periods of the order of hours , as opposed to circadian " 24 hour rhythms .we will discuss specific details of each system , at the same time emphasising that the basic physical mechanism which produces oscillations in all three is the same negative feedback along with time delays .the protein p53 is responsible for inducing apoptosis in cells with damaged dna .the concentration of p53 is usually kept low by a feedback mechanism involving another protein , mdm2 , which binds to p53 and promotes its degradation . when the dna is damaged , the cell expresses a number of kinases which phosphorylate ser20 in p53 , changing its affinity to mdm2 .this results in oscillations in the concentration of p53 , observed both in western blot analysis ( cf .[ fig_oscill](a ) ) and in single cell fluorescence experiments .the standard explanation for the overall increase in the concentration of p53 is that its phoshoprylation decreases its affinity to mdm2 , shifting the thermodynamic equilibrium towards higher concentrations .apart from not explaining the oscillations , this argument does not agree with other experimental evidence .equilibrium isothermal titration calorimetry experiments have shown that phosphorylation at ser20 _ decreases _ , not increases , the dissociation constant between p53 and mdm2 from to .the same effect is observed _ in vivo _ , where p53asp20 ( a mutated form which mimics phosphorylated p53 ) binds mdm2 more tightly than p53ala20 ( which mimics unphosphorylated p53 ) .moreover , single cell experiments show a slight decrease in the concentration of p53 after dna damage , which can not be explained by the standard argument .we showed that a simple model of the p53-mdm2 system that incorporates the time delay associated with some relatively slow processes within the cell can account for the experimental facts in a simple way .the feedback mechanism is sketched in fig . [ fig_sketch](a ) and the associated time - delayed rate equations are where the delay takes into account the half - life of mrna , the diffusion time , the time needed to cross the nuclear membrane and the transcription / translation time .one can solve eqs .( [ eq_p53 ] ) numerically , simulating the damage to dna by a sudden change in the dissociation constant .[ fig_p53o ] shows the onset of oscillations in this model in response to a sudden decrease in at time .  , , , , , , and .at time s the value of is decreased of a factor 10 . in the inset , the solution of the same equations , where the value of is increased of a factor 10 at s. the equations are solved with the adams algorithm . ,width=377 ]  several observations emerge from the simulation results in fig .[ fig_p53o ] .most interestingly , what triggers oscillations is a decrease in the dissociation constant , while any increase in its value just shifts the equilibrium towards higher values of ( cf . inset of fig .[ fig_p53o ] ) .moreover , just after the concentration of p53 decreases , as observed in the experiments . finally , the concentration , , of the complex p53-mdm2 is , at any time , essentially identical to the minimum of and ( gray curve in fig .[ fig_p53o ] ) , indicating that the complex is saturated .  the relative change , , in the maximum concentration of p53 reached after the simulated stress event is displayed in fig .[ fig_p53k ] as a function of the change in .the value of displays a sigmoidal behaviour if the stress decreases the value of .in contrast , in the absence of a time delay , this curve is linear with respect to  .thus , the time delay and the oscillations are crucial for producing a sensitive system that can respond to stress in a sharp , faster - than - linear manner .we also undertook a detailed analysis of the response of the oscillations to changes in the parameters .it turns out that the oscillations do not change qualitatively when parameters , and are varied ( by upto five orders of magnitude ) , whereas a decrease in or suppresses the oscillating behaviour .these parameters are associated with , respectively , the the degradation of mdm2 and the binding of p53 to the dna .one could speculate therefore that these processes play an important role in the production of tumors that arise when the p53 response mechanism fails .this speculation is , in fact , consistent with the observation that around the 45% of all tumors display mutations in the p53 region which binds to dna .the overall behaviour of the proteins with respect to time evidently depends on the parameters of the dynamic equations , that is the production and degradation rates and the delay . over long timescales, the protein concentrations can either converge to a steady state or a limit cycle ( i.e. , sustained oscillations ) .chaotic behaviour is never observed within this model .a careful analysis of how the dynamics of the protein concentration depends on the parameters of the rate equations has been done by neamtu and coworkers in ref .the main conclusion of this work is that under the condition that the dissociation constant between p53 and mdm2 is small , there exists a critical delay , , above which the system shows oscillations ( see details in appendix c ) .  in the language of dynamical systems ,the transition when the delay , , crosses is a hopf bifurcation .a hopf bifurcation is the generic type of bifurcation that occurs when a stable fixed point of the system becomes unstable and turns into a limit cycle , a dynamically oscillating state , as consequence of a change in some parameter of the system ( see appendix b for more details ) .the transcription factor hes1 controls the differentiation of neurons in mammalian embryos .its concentration is controlled by a feedback loop built out of hes1 and its own mrna ( see fig . [ fig_sketch ] ) . like p53 , its concentrations displays oscillations when the cells are stimulated with serum .the period is similar to that of p53 , approximately two hours , and the oscillations last for hours .hes1 is known to repress another protein called mash1 .both the knocking out of mash1 and its continuous expression by means of retroviral introduction results in a lack of cell differentiation .only when there are periodic oscillations in the concentration of hes1 ( and thus of mash1 ) is there proper differentiation of neuronal cells , showing that the temporal variation of the protein concentrations are critical for the developement of the nervous system .the relation of hes1 oscillations to segmentation and spatial patterns has been studied by several authors .it is well known that oscillations in hes1 is part of notch signalling and hirata et al ref . studied the coordinated somite segmentation in the presomitic mesoderm of mice embryos and found a correlation between the oscillations of hes1 and initiation of somites .the general issue of segmentation in vertebrates has been further studied in ref . indicating that the oscillations in the notch pathway signaling are intimately related to the wnt signaling . at the caudal end of the embryo notch and wntoscillates out of phase when the gradient in the wnt level is over a given threshold .new cells are provided and move away from the caudal end each setting a boundary for a somite segmentation .this continues until all somites are produced which subsequently gives rise to the spinal cord .ultradian oscillations of signaling pathways are thus extremely important for segmentation .the control mechanism of hes1 oscillations again involves a negative feedback loop with one activation and one repression : the transcription of the mrna of hes1 activates the production ( translation ) of the protein , and hes1 represses the transcription of its own mrna .the main difference compared to p53 is that here one of the nodes represents an mrna , not a protein .however , this difference is merely semantic : the control network still has two nodes , of which one is activating and one is repressing .the molecular species of these nodes are immaterial to the description of the oscillations .we model the system using the following equations ( cf . ) ^h } -k_r r(t)\\ \frac{\partial s}{\partial t } & = & \beta r(t ) - k_s s(t ) , \label{eq_hes1 } \end{aligned}\ ] ] where and are the concentrations of hes1 and its mrna , respectively .the meaning of these equations is that mrna is produced at a rate when hes1 is bound to the dna .the probability that hes1 is bound to dna is , where is a characteristic concentration for dissociation of hes1 from the dna , and is the hill coefficient that takes into account the cooperative character of the binding process . and are the spontaneous degradation rates of the two proteins , while is the delay associated with the molecular processes that we do not want to describe explicitly ( transcription , translocation , etc . ) . ref . suggests that and are of the order of 25 minutes .the value of the time delay is difficult to assess , since it is determined by a combination of various molecular processes .one can guess that its order of magnitude is tens of minutes .  ) ., width=377 ]   of hes1 as a function of the delay ., width=377 ]  the numerical solution of eq .( [ eq_hes1 ] ) is displayed in fig .[ fig_hes1 ] .the oscillations have a period min .the dependence of the time period on the delay is shown in fig .[ fig_hes1del ] . for any delay in the range min ,the oscillation period is consistent with that found experimentally , and so is the time difference between the peaks in hes1 and mrna , min . for min ,the system shows no oscillations . to check the robustness of the results , we have varied , and over 5 orders of magnitude around the basal values , listed in the caption to fig .[ fig_hes1 ] , and observed no qualitative changes to the oscillatory behaviour . on the other hand ,a decrease of and disrupts the oscillations .this is because these two quantities set the timescale of the dynamics , increasing this timescale , keeping constant , is equivalent to decreasing and thus no oscillations are produced .the behaviour of hes1 is very similar to that of p53 , both in the features of the oscillations and in the lag delay before they start .this is not unexpected , since the structure of eq .( [ eq_hes1 ] ) is very similar to the structure of eq .( [ eq_p53 ] ) and to any time delay equation describing a two component negative feedback loop .the nf- family of proteins is one of the most studied in the last ten years , being involved in a variety of cellular processes including immune response , inflammation , and development .nf- can be activated by a number of external stimuli including bacteria , viruses and various stresses and proteins ( for instance , refs . used the tumor necrosis factor- , tnf- ) . in response to these signalsit targets over 150 genes including many chemokines , immunoreceptors , stress reponse genes , as well as acute phase inflammation response proteins .each nf- has a partner inhibitor called i , which inactivates nf- by sequestering it both in the nucleus as well as in the cytoplasm .in fact , the i proteins come in several isoforms  , and perhaps others .some of these isoforms are , in turn , transcriptionally activated by nf- , thus forming a negative feedback loop which is essentially identical in structure to the other two discussed above .the potential for this negative feedback loop to produce oscillations in the nuclear - cytoplasmic translocation of nf- was initially shown by electrophoretic mobility shift assay experiments .they found that wild - type cells and mutants containing only the i isoform showed damped oscillations .in contrast , cells with only the i or isoforms do not show oscillations .this conclusion was bolstered by single - cell flourescence imaging experiments which show sustained oscillations of nuclear nf- in mammalian cells , with a time period of the order of hours . in these experimentsi was overexpressed hence the system behaves like the mutant which has only the i isoform .green arrows indicate transcription and translation .blue arrows indicate transport processes .purple double arrows indicate complex formation .the red dashed arrow indicates ikk triggered degradation of i when complexed to nf- .[ nfkbschematic],width=377 ]  as sustained oscillations are clearest here , we will focus our modelling on this mutant .the following cellular processes , summarized in fig .[ nfkbschematic ] , are important for this system on the timescales we are interested in ( i.e. , we ignore processes which are very slow ) : +  * nf- , when in the nucleus , activates transcription of the i gene ( henceforth we will drop unless we are explicitly talking about more than one isoform ) , producing i mrna in the cytoplasm .* the i mrna is translated to form i protein .* the i protein can be transported in and out of the nucleus . * in both compartments , i forms a complex with nf- . *the nf- - i complex ( henceforth referred to as \{ni } ) can not be imported into the nucleus .however , if it forms within the nucleus it can be exported out .* free nf- behaves in exactly the opposite way .free nf- is actively transported into the nucleus but not from the nucleus to the cytoplasm . *the cytoplasmic \{ni } complex is tagged by another protein , the i kinase ( ikk ) , for proteolytic degradation .this results in degradation of i only , releasing nf- .note that this degradation does not occur for free i .  on the timescales of interest, there is no net production or degradation of nf- .it simply cycles in and out of the nucleus , i.e. , the sum of nuclear and cytoplasmic nf- concentrations is a constant . amongst the above listed processes, the association and dissociation of the complex \{ni } occurs fast enough that the concentration of the complex can be taken to be always in equilibrium with the free nf- and i concentrations .this allows us to describe the system using a very simple model consisting of only three variables , nuclear nf- ( ) , cytoplasmic i ( ) and i mrna ( ) :   the terms in equation ( [ nfkb1 ] ) model the nuclear import and export of nf- .( [ nfkb2 ] ) models nf- - activated transcription of the i gene and spontaneous degradation of the mrna . finally , eq . ( [ nfkb3 ] ) has terms for translation of i mrna into the protein and ikk mediated degradation of i .the external signal is supplied by ikk that enters the equations through the parameter , , which is proportional to ikk concentration .b ( ) , red , and cytoplasmic i , green , for , , , and ( these parameter values are derived from the ones used in ref . , see . ) in order to facilitate comparison with the experimental plot , the x - axis has been limited to 600 minutes , but the oscillations are in fact sustained ( see fig .[ spikyandsoft]a).,width=377 ]  these equations produce sustained oscillations in all variables over a wide range of parameter values . fig .[ osc ] shows the result of simulations which use parameter values from ref .the oscillations produced by this model match the experimentally observed features well , in particular , the shape , phase , time period and frequency response are correctly reproduced .note that this is certainly not the only model able to reproduce the experimental oscillations observed in nf- .et al . _ have constructed a long list of chemical reactions between 26 different molecules in the nf- system , including 65 numerical parameters .the three variable model above was formed by reducing this larger model , also producing a 7-variable and 4-variable model in the process .hayot and jayaprakash have also built a model with seven variables for nf- oscillations .another model , which is similar to hoffman et al s model , but has an additional feedback loop is described in .a key element in the model is the saturated degradation of cytoplasmic i in the presence of ikk ( second term in eq .( [ nfkb3 ] ) ) .this saturation occurs because the level of the nf- - i complex saturates , and this complex is needed for ikk triggered degradation of i .a stability analysis of the system shows the importance of the saturated degradation for oscillations .we begin by examining the fixed points of the system .the fixed point values of , and are solutions to    and can be eliminated using giving from this we find that the fixed point value of is a solution of the equation , ] .notice that the hopf theorem does not ensure that the orbit is stable ; however , since the system is bounded and there are no other fixed points , we expect the orbit to be attracting , at least close to the transition point .notice that condition ( [ stabcond ] ) is always satisfied when .this result also extends to the more general case where degradation rates are unequal .thus , we have proven that a two - component monotone negative feedback loop without an explicit time delay can never show oscillations .usually , the bifurcation takes place under the variation of some external control parameter and the new state appears at a critical value of this parameter . for genetically regulated networs, the control parameter could be some external chemical stimuli , a production rate , or a binding constant , just to mention a few examples . to illustrate a hopf bifurcationlet us first consider a completely general two - dimentional dynamical system ( without delay ) defined by two coupled non - linear differential equations ( we use variables to resemble p53 and mdm2 ) .the stationary fixed point is determined by using standard routines one linearizes around the fixed point by means of the jacobian matrix where the star symbolizes that we insert the fixed point into the jacobian .the resulting set of eigenvalues can either both be real or be a set of two complex conjugates .this is the case we are interested in here : for ( correpsonding to ) the fixed point is stable whereas the limit cycle becomes stable for ( corresponding to ) and performing oscillations on the limit cycle with a frequency defining th period of the oscillation . for a genetic systemthis period could for instance be circadian , ultradiand or related to cell cycles . just above the bifurcation, the limit cycle can be well approximated by a circle whoes radius for super - critical hopf bifurcations generally grows continuosly as ( as in contrast to a sub - critical hopf bifurcation where the radius will jump and exhibit hysteresis effects ) .when the control parameter is much larger than the limit cycle may easily be deformed in various ways away from the circle .when we introduce a time delay in the equations , as discussed above , we can formally write it by introducing the delayed variable into the equations : we find numerically that the system still exhibits hopf bifurcations when crossing critical lines in the parameter space .in fact , we find that one can drive the system through hopf bifurcations by increasing the value of the time delay through specific values .the fixed point of is of course equal to the fixed point of , i .. e . : .now we use the same approach as above and linearize around the fixed point : which leads to the following dynamical equations for the increments :  we can now assume solutions on the form : and find after some lengthy algebra the eigenvalues : \ ] ] \ ] ] we note that these are transcendal equations in the eigenvalues but that the time delay specifically appears into the relations ( for a further discussion , see previous appendix ) .  in many biological systems with genetic feed back regulations , there are often more than two dynamical variables .we shall in the following sections discuss oscillatory states of the transcription factor nf- .as we show , that system can favourably be reduced to a three dimensional dynamical system in the variables nf- , its inhibitor i and the associated mrna , i .let us for simplicity write these variables as thus formally obtaining the following three dimensional dynamical system : the procedure to study this type of quations is exactly the same as for the two - dimensional system .from the stationary point we linearize around it using the jacobian matrix . in this case one obtains three eigenvalues which are either all three real , or one is real and the two others complex conjugates . again , in the last case one may encounter a hopf bifurcation where the fixed point bifurcates into a limit cycle in the three - dimensional space , causing oscillations in these variables .notice , that there topologically is a big difference between oscillatory limit cycles in two and three dimensions . in two dimensionsa limit cycle can enclose trajectories , which is not the case in three dimensions where tracjectories may escape around the limit cycles .an important theorem for this type of behavior is called the poincare - bendixson theorem , which states that if a trajectory is confined to a closed , bounded region and there are no fixed points in that region , then the trajectory eventually approaches a closed orbit ( see e.g. ) .consider the simplest delayed system of the form a complete description of the solution is given in an appendix of ref . , while we will just investigate the conditions where the system display sustained oscillations .the general solution of eq .( [ eq_app_linear ] ) is which , inserted in t investigate the conditions where the system display sustained oscillations .the general solution of eq .( [ eq_app_linear ] ) gives defining , the oscillatory case takes place when .the solution is then which gives a more general treatment gives the conditions under which the system converges to a stationary solution , that is for any or if for . roughly speaking, the system can oscillate if the dominant part of the kernel is the delayed one and if the delay is large enough .if we label the vector forming the left hand of eq .( [ general ] ) as and that forming the side right hand side as , its divergence of is since we exclude that each of the two molecules can activate itself , the partial derivatives and are non positive , and consequently . by virtue of green s theorem ,if the trajectory were closed , where is the normal to the trajectory , and consequently everywhere .since the circulation of a null vector is zero , this leads to a contradiction , and the trajectory can not be closed .neaumtu and coworkers develop in ref . the complete stability analysis of the p53-mdm2 system with delay .first they show that for any choice of the parameters there is a unique stationary point for the rate equations .consider the system obtained by linearization of eq .( [ eq_p53 ] ) aroud such point where and are the derivatives of with respect to and in the stationary point , while and are the derivatives of the function .define , , and .let be the eigenvalues of the linearized system .it is possible to prove that if and then there is a delay such that and consequently a hopf bifurcation occurs at the stationary point .geva - zatorsky n , rosenfeld n , itzkovitz s , milo r , sigal a , dekel e , yarnitzky t , polak p , liron y , kam z , lahav g , alon u 2006 oscillations and variability in the p53 system _ mol .* 2 * 0033  nelson d e , ihekwaba a e c , elliott m , johnson j r , gibney c a , foreman b e , nelson g , see v , horton c a , spiller d g __ 2004 oscillations in nf - kb signaling control the dynamics of gene expression _ science _ * 306 * , 704708  bosisio d , marazzi i , agresti a , shimizu n , bianchi m e and natoli g 2006 a hyper - dynamic equilibrium between promoter - bound and nucleoplasmic dimers controls nf - kappab - dependent gene activity _ embo j. _ * 25 * , 798810 .metivier r , penot g , hubner m r , reid g , brand h , kos m and gannon f 2003 estrogen receptor - alpha directs ordered , cyclical , and combinatorial recruitment of cofactors on a natural target promoter _ cell _ * 115 * 751763 .kucho k , okamoto k , tsuchiya y , nomura s , nango m , kanehisa m and ishiura m 2005 global analysis of circadian expression in the cyanobacterium synechoscystis sp .strain pcc 6803 , _j bacteriol _ * 187 * 2190 - 2199 .basak s , kim h , kearns jd , tergaonkar v , odea e , werner s l , benedict c a , ware cf , ghosh g , verma i m , hoffmann a 2007 a fourth ikappab protein within the nf - kappab signaling module . _ cell . _ * 128 * 369381   kearns j d , basak s , werner s l , huang c s , hoffmann a 2006 ikbe provides negative feedback to control nf- oscillations , signaling dynamics , and inflammatory gene expression _ j. cell biol . _ * 173 * 659 - 664        ishiura m , kutsuna s , aoki s , iwasaki h , andersson cr , tanabe a , golden ss , johnson ch , kondo t 1998 expression of a gene cluster kaiabc as a circadian feedback process in cyanobacteria _ science _ * 281 * 15191523    jacquet h , renault g , lallet s , mey j d and goldbeter a 2003 oscillatory nucleocytoplasmic shuttling of the general stress response transcriptional activators msn2 and msn4 in saccharomyces cerevisiae _ j. cell biol ._ * 161 * 497505    goldbeter a , dupont g and berridge m 1990 minimal model for signal - induced ca2 + oscillations and for their frequency encoding through protein phosphorylation _ proc . natl .* 87 * 14611465

Abstract: the development of new techniques to quantitatively measure gene expression in cells has shed light on a number of systems that display oscillations in protein concentration . here we review the different mechanisms which can produce oscillations in gene expression or protein concentration , using a framework of simple mathematical models . we focus on three eukaryotic genetic regulatory networks which show ultradian " oscillations , with time period of the order of hours , and involve , respectively , proteins important for development ( hes1 ) , apoptosis ( p53 ) and immune response ( nf- ) . we argue that underlying all three is a common design consisting of a negative feedback loop with time delay which is responsible for the oscillatory behaviour .


several authors in the physics community ( including , , , , , , , and ) have written about the appearance of log - periodic oscillations in stock market indices during the months and years leading up to financial crashes . by a log - periodic oscillation , we mean that a variable exhibits a functional form such as  \text{.}\label{lp}%\ ] ] the cosine term in this expression produces oscillations with a period that grows or shrinks exponentially .as an example , if we let , where is the time of the well - known crash of october 19 , 1987 , then we can see this pattern in the standard and poor s 500 ( s&p 500 ) during the five years prior to the crash as shown in fig .[ abs index ] .such log - periodic oscillations are of interest to physicists because log - periodicity is the signature of a spatial environment with a discrete scaling symmetry . the observation of log - periodic precursors has been interpreted as evidence that the subsequent financial crash can be viewed as a critical point analogous to a phase transition in a more traditional physics environment .heuristically , this picture of a transition to an ordered state is certainly consistent with what happens during a stock market crash , when virtually all investors will be in agreement that the price is dropping and that they should sell , in contrast to the normal disordered state where the price remains fairly stable because there is a buyer for every seller .the theoretical issue is how far this analogy can properly be taken .[ ptb ]  lpfig1.epsi  one can envision a discrete scaling symmetry arising in financial markets if one models investors as forming networks of friends and contacts .if every agent has contacts , it is indeed possible to construct a network which exhibits a scaling symmetry .johansen et al . propose a model of imitative investors , living in such a network , which is essentially an ising spin model .a spin - up constituent represents a buyer , a spin - down constituent represents a seller , and neighboring spins interact in a manner that encourages them to align in the same direction .johansen et al .respond to the criticism that a rational agent could exploit log - periodic fluctuations to his financial advantage , and thereby acquire all the wealth in the economy , by requiring that stock prices obey a martingale condition that would rule out arbitrage opportunities .how the collective behavior of the imitative investors translates into a price that satisfies this martingale condition is left unmodeled .regardless of the underlying details , this martingale condition is a very strong restriction on the behavior of prices , and testing this condition is the primary focus of the present paper .a major deficiency of the vast body of previous work on log - periodic precursors is that most of the analysis has been qualitative rather than statistical in nature .the emphasis has been placed on establishing the existence of log - periodic fluctuations , usually through curve - fitting or more recently through the use of lomb periodograms ( ) .we should note that we take it as a given in this paper that such methods are valid .we do not dispute the existence of log - periodic fluctuations like those pictured in fig .[ abs index ] .what remains controversial is why these fluctuations exist .are they indeed part of a collective process leading up to a phase transition ? or can they be more simply explained as an accidental feature of the stochastic processes that drive financial indices ? only statistical methods have the power to distinguish these possibilities .a small amount of work has already been done in this vein .feigenbaum and freund modeled the stock market as a random walk and found that the average time between a log - periodic spell and the next financial crash was significantly longer in real - world data than in monte carlo simulations .johansen et al .( , ) report similar results from an experiment where they modeled the stock market as a garch process .while these are positive results with respect to the hypothesis of crashes as critical points , they are both strongly tied to a specific model of the underlying stochastic process , what statisticians term the data generating process ( dgp ) .one can not extrapolate from these two results to conclude that no dgp can account for log - periodicity .the null hypothesis that log - periodic spells are the casual products of random fluctuations.has _ not _ been rejected .while most studies of log - periodicity have focused on prices , the present paper applies statistical hypothesis testing to the question of whether the first differences of a price process exhibiting log - periodic fluctuations are also log - periodic .this question is of interest because any model in which log - periodic price fluctuations reflect corresponding fluctuations in the mean of the price process will require that the mean of the first differences should also behave log - periodically . in particular , the model of johansen requires this .moreover , it is well - known among econometricians that regression estimators are inconsistent when applied to a serially - correlated time series like a stock price . first differencing the stock pricecan substantially eliminate this serial correlation and may eliminate effects produced by it .consequently , the absence of log - periodic fluctuations from the first differences of a price process suggests ( although certainly does not prove ) that log - periodicity in the integrated price process may be due to serial correlation and random effects .examining the first differences for the logarithm of the s&p 500 from 1980 to 1987 , we find that the log - periodic component of the sornette - johansen specification is statistically significant if we include this entire period in the fit .however , if we exclude the last year , the log - periodic component is no longer statistically significant .furthermore , the best fit in this truncated data set predicts a crash in june of 1986 , shortly after the data set ends but well before the actual crash .presumably the log - periodic fit is coupling to some structure in the data just prior to the crash which does not characterize the whole time series .thus we conclude that log - periodicity is either negligible or not present in these first differences , a negative result for the model of johansen et al . .  a second claim that we consider was put forward by johansen and sornette and pertains to the distribution of drawdowns , where a drawdown is the accumulated drop in an index from a local maximum to the next local minimum .assuming that the fat - tailed distribution of daily returns falls off exponentially in the limit of large negative returns , they derive that the distribution of drawdowns should also be exponential .empirically , they find that the frequency distribution of drawdowns over the past century does indeed look exponential except for three outliers , the 29 and 87 crashes and a crash during world war i. they infer from this that a different mechanism must be responsible for these outliers and suggest that this mechanism is also responsible for producing log - periodic oscillations . in principle , however , one can examine any set of data with one dependent variable and say that the graph looks exponential except for some outliers . in order to make this a testable claim ,one has to consider whether there are statistically significant deviations from the exponential curve .in fact , statistically significant deviations do occur , and we conclude that this evidence of a special crash mechanism is unconvincing .  the paper proceeds as follows .in section [ sjp ] , we review the basic elements of the sornette - johansen framework and note some theoretical criticisms .since the initial papers on this subject appeared , sornette and johansen have developed a more sophisticated specification for observed log - periodic oscillations , treating the logarithm of the price index as the dependent variable rather than the price itself .we will see in section [ logspec ] that the logarithmic specification does fit the s&p 500 prior to the 87 crash .it also fits to the s&p 500 prior to downturns in 1974 , 1997 , 1998 , and an otherwise unremarkable downturn of less than 1% in 1985 .this last finding raises the spectre that perhaps log - periodic behavior is not especially noteworthy and is not inherently related to crashes . in section [ fd] , we examine the first differences of the logarithm of the s&p 500 during the period of 1980 to 1987 and test the statistical significance of the log - periodic component of the sornette - johansen specification . in section [ ddd ] , we consider the distribution of drawdowns over the period from 1962 to 1998 and examine whether the 87 crash deserves classification as an outlier. finally , in section [ cr ] , we offer some concluding remarks .johansen et al . model the behavior of a speculative stock which pays no dividends . they allow for the possibility that at least one rational , risk neutral agent with rational expectations , who does not discount the future , exists in their economy .given this assumption , the stock price must follow a martingale . for all ,  = p(t),\label{martingale}%\ ] ] where denotes the expectation operator conditional on all information available at .strictly speaking , the operator in eq .( [ martingale ] ) would refer to an expectation based on the beliefs of the hypothesized agent .when we say the agent has rational expectations , we mean the agent s beliefs in fact correspond to reality , so is also the objective expectation operator .  in a market equilibrium where agents behave optimallyaccording to their preferences , eq .( [ martingale ] ) is an endogenous no - arbitrage condition that must hold for markets to clear . if >p(t) ] , rational agents will find it profitable to sell short an infinite amount of the stock .since the supply of stock is finite , neither of these two possibilities is consistent with equilibrium .since the stock pays no dividends , its fundamental value ( the expected present value of the stream of future dividends ) is zero , so any positive price constitutes a `` bubble '' . in a world of perfect certainty , for , eq .( [ martingale ] ) would translate to the trivial difference equation and it would follow by mathematical induction that must be a constant .if we assume that bubbles can not be sustained forever , then a transversality condition ( essentially the boundary condition at the end of time ) will hold : in that case , we will have for all .given the transverality condition , bubbles can not arise under perfect certainty . with uncertainty, however , eq . ( [ martingale ] ) is no longer a deterministic difference equation and will have an infinite number of solutions that satisfy the generalized transverality condition =0\text{.}% \label{uncer lim}%\ ] ] ( see chapter 5 of for a review of bubble solutions . )  consequently , johansen et al . consider a price process which satisfies eq .( [ martingale ] ) but also introduces the possibility of a crash .let denote a random variable that equals 0 before the crash and 1 afterwards. denote the continuous distribution function ( cdf ) of the crash time as and the corresponding probability density function ( pdf ) as .the hazard rate , the probability per unit time that the crash will happen in the next instant if it has not already happened , will then be if we assume that the crash involves a downturn of a fixed percentage of the price , then the price process can be described as where is a mean - zero noise term . the time - dependent drift is then determined by eq .( [ martingale ] ) : =\mu(t)p(t)dt-\kappa p(t)h(t)dt=0\text{.}%\ ] ] thus disregarding the noise term ( whose contribution may be computed using ito calculus ) , eq .( [ sde ] ) has the solution  = \kappa\int_{t_{0}}^{t}h(t^{\prime } ) dt^{\prime}\label{price}%\ ] ] prior to the crash .the connection of this theory to physics lies in the determination of the hazard rate function .in addition to any rational agents in the economy , johansen et al . posit the existence of a large number of irrational noise traders , and it is these noise traders who are responsible for any dynamics leading up to a crash .they construct a `` microscopic '' model in which the noise traders are imitative investors who reside on a local interaction network .neighbors of an agent on this network can be viewed as the agent s friends or contacts , and an agent will incorporate the views of his neighbors regarding the stock into his own view .  at time , each agent is assumed to be in one of two states : a bullish state or a bearish state .if we denote the set of nearest neighbors as , given the states of all his neighbors at , s state at will be determined by where is a constant and is an i.i.d .random variable with =0 ] , and this would be correct if the pricing kernel was a constant . however , the pricing kernel in general is not a constant and is not even a function of the price .thus , it is not obvious that variations in the pricing kernel could not counteract variations in the price in such a way as to disrupt any log - periodicity .moreover , the determination of the pricing kernel is a highly non - trivial problem if there exists a multiplicity of rational agents who are not identical . since johansen et al . do not model how the collective behavior of the noise traders is translated into a price ,it is not possible for us to even begin such a computation .but corresponding to every rational agent , there would be an equation like . ( [ gen mar2 ] ) which would have to be satisfied , andit is not clear that a log - periodic price trend could be made compatible with each and every one of these martingale conditions .nevertheless , if we ignore the possibility of risk - averse agents , we do have a workable model , and we will see in section [ fd ] whether eq .( [ price ] ) with a log - periodic hazard rate function is consistent with the behavior of the s&p 500 prior to the 1987 crash .although early investigations of log - periodic fluctuations in stock market indices prior to financial crashes focused on the specification of eq .( [ lp ] ) , sornette and johansen introduced an alternative specification in which it is the logarithm of the price rather than the absolute price which fluctuates log - periodically .this logarithmic specification is preferable to eq .( [ lp ] ) because it is widely assumed that investors are primarily concerned with relative changes in stock prices rather than absolute changes in eq .( [ u ] ) is either or for ) . ] .it is also more straightforward to compare the logarithmic specification to the once popular hypothesis that the logarithm of a stock price will follow a random walk with drift .let be a time prior to the crash at which the stock price has begun to fluctuate log - periodically .we define and  .\ ] ] then for ,  = a+bf_{1}% ( t)+cf_{2}(t),\label{logps}%\ ] ] where is the objective expectation operator at and if no crash has occurred at or prior to .this specification has nine free parameters : , , , , , , ,  , and .note that in the sornette - johansen paradigm , does not necessarily correspond to the time of the crash per se ( which we will denote by ) but to a critical time where the probability of a crash becomes very large . in terms of the microscopic model in , where is the number of nearest neighbors of each agent .consequently , if their model is correct , .sornette and johansen also suggest making this restriction so that does not become unbounded as .however since log - periodic behavior typically breaks down as gets close to , it has become standard practice to disregard data points very close to the observed crash , in which case eq .( [ logps ] ) will be bounded in the time span of interest even for .the canonical example of a log - periodic precursor can be seen during the years prior to the october 1987 crash as shown in fig .[ abs index ] , so we first consider the closing value of the s&p 500 for each day between january 1 , 1980 and september 30 , 1987 .suppose there are trading days in this period . given a choice of parameters , , , , , , , , and , we define an -dimensional residual vector by where is the time of the trading day . the standard practice for estimating the parameters of eq .( [ logps ] ) is to use the least squares estimation procedure , choosing those values which minimize the square of the residual vector .econometricians define the coefficient of determination where and is the sample mean of .the coefficient of determination is often interpreted as the ratio of the amount of variation in the dependent variable explained by a model to the total amount of variation in that dependent variable .the closer is to unity , the more variation in the dependent variable is explained by the model .note however that is not invariant to linear transformations of the , so one should not put too much emphasis on its value .notwithstanding , one could equally well characterize the least squares procedure as maximizing the coefficient of determination .( [ logps ] ) is linear in the parameters , , and and nonlinear in the remaining parameters .we use the downhill simplex method ( see chapter 10.4 of ) to minimize as a function of , , , , , and . for each choice of these nonlinear parameters ,we obtain the best values of the linear parameters , , and using ordinary least squares ( ols ) methods .the square of the residual vector is not a well - behaved concave ( down ) function , so the local minimum obtained by the downhill simplex method will depend on the initial starting point fed to the routine .indeed , as was noted in , one typically finds many local minima with fairly similar values of .nonlinear least squares ( nlls ) estimation requires that as the sample size goes to infinity , should converge to a function with a unique global minimum in some chosen parameter space ( see ) .however , nlls theory does not require that have a unique global minimum within the parameter space for finite samples , and the theory does not offer much guidance about how to deal with a multiplicity of local minima having similar values that deviate within sampling error .the minimum with the lowest observed value need not be the closest minimum to the truth .the first and second columns of table [ parval ] describe two such local minima for the years prior to the 87 crash .the corresponding fits are plotted in figs .[ 8087b.7 ] and [ sp8087b1 ] .notice that in the fit of fig .[ 8087b.7 ] , in accord with the microscopic model of .however , the fit of fig . [ sp8087b1 ] has , and the square of the residual vector is a bit smaller in this second fit .in addition to the multiplicity issue , we face another difficulty because there is really no question that the are serially correlated , which complicates the estimation of standard errors even for the linear parameters , , and . as a result , we do not report standard errors in table [ parval ] , and we will postpone a discussion of whether the individual terms in eq .( [ logps ] ) are statistically significant until the next section , where we consider the first differences of the price index .[ c]ccccccc & 10/19/87 & 10/19/87 & 10/3/74 & 12/2/85 & 10/27/97 & 8/31/98 + & 1959 & 1959 & 342 & 365 & 397 & 352 + & 1/2/80 & 1/2/80 & 4/26/73 & 5/23/84 & 3/7/96 & 3/11/97 + & 9/30/87 & 9/30/87 & 8/30/74 & 10/31/85 & 9/30/97 & 7/31/98 + & 11/14/87 & 9/30/87 & 9/22/74 & 11/14/85 & 1/3/98 & 3/2/99 + & 5.89 & 5.77 & 2.84 & 5.24 & 7.07 & 7.29 + & -.0046 & -.00041 & 1.13 & & -.0066 & -.0024 + & .00051 & -.000056 & .0199 & & -.00042 & .00022 + & 0.74 & 1.06 & 0.10 & 3.53 & .73 & .84 + & 9.64 & -2.24 & -1.20 & 14.64 & 4.95 & 10.41 + & -4.34 & -0.38 & -0.02 & -6.47 & 1.60 & -10.58 + & 8.39 yr & 4.84 yr & 489.63 yr & 0.98 yr & 3.18 yr & 12.74 yr + & 11.65 & 28.85 & 55.90 & -0.30 & 31.63 & 72.44 + & 4.02 & 3.36 & .24 & .12 & .18 & 0.15 + & .978 & .981 & .932 & .944 & .973 & .969  [ ptb ]  lpfig2.epsi  [ ptb ]  lpfig3.epsi  [ ptb ]  lpfig4.epsi  [ ptb ]  lpfig5.epsi  [ ptb ]  lpfig6.epsi  [ ptb ]  lpfig7.epsi  for now , we will simply observe that it is possible to obtain a `` reasonable '' fit of eq .( [ logps ] ) to the s&p 500 between 1980 and 1987 . between 1962 and 1998, the next three most significant drawdowns occurred in october 1974 , october 1997 , and august 1998 . as we demonstrate in figs .[ sp7374]-[sp9798 ] , `` reasonable '' fits ( as reasonable as the fits in for example ) to eq .( [ logps ] ) can also be obtained during the year prior to each of these drops .this evidence might lead one to conclude that there must be a correlation between spells of log - periodic fluctuation and financial crashes .however , this would be fallacious reasoning because we have not demonstrated that such spells do not occur ( with significant frequency ) during periods that do not culminate in a financial crash .in fact , when we randomly selected a drawdown from the list of drawdowns between 1962 and 1998 with total magnitude on the order of 1% , we picked a drop of 0.84% that occurred at the beginning of december 1985 .we display the s&p 500 during the year prior to this small drop in fig .[ sp8485 ] , and we see that this also can be fit to eq .( [ logps ] ) .this suggests the possibility that there is nothing particularly special about a finding of log - periodic fluctuations in the stock market .laloux et al . also relate an experience where they observed log - periodic fluctuations in a financial index , predicted a crash , and then were disappointed when none occurred .the hypothesis of a correlation may still be rescued , however , if it can be shown that fits with parameter values in a select set are harbingers of crashes .this was the approach taken by feigenbaum and freund , although their definition of a log - periodic spell was ad hoc and not motivated by any statistical theory . between 1980 and 1994, they found three independent periods that satisfied their definition .a monte carlo simulation extending over 10,000 days , which modeled the stock market as a random walk , produced 17 such spells .the average time between a log - periodic spell and the next crash was substantially less in the empirical data than it was in the simulation , which suggests that there might indeed be an association between log - periodic crashes and financial crashes .it should be noted that they worked with the specification of eq .( [ lp ] ) , which has fewer parameters than the specification of eq .( [ logps ] ) .it is possible that adding these extra parameters makes `` good '' fits to eq .( [ logps ] ) less extraordinary .johansen et al .( , ) describe a similar experiment where they modeled the stock market as a garch process and searched for fits which extended for about eight years and had , , and parameters that fell within a specific interval .they also did not find any linkage between a log - periodic spell and an ensuing crash .another test on the statistical significance of log - periodicity was done by huang et al . in the context of earthquakes .they studied the lomb periodograms of monte carlo simulations generated by a dgp with normal i.i.d .noise in order to assess the statistical significance of observed periodograms .unfortunately , all of these results are weakened by the fact that only one possible data generating process was ruled out by each experiment .furthermore , the dgps used in the two finance experiments were chosen for their simplicity , not because they are credible models of the actual dgp for a stock price .feigenbaum and freund used a random walk , but there is really no question any more in the finance community that stock prices do not follow random walks .as for garch processes , while these are considered a much better model of the stock market during ordinary periods , there is no reason to believe they are an appropriate model of the stochastic process during an extreme event like a financial crash .in fact , the frequency of crashes in johansen et al.s monte carlo simulations was much smaller than the frequency of crashes in real data , so their dgp obviously does not adequately capture the behavior of stock prices during and leading up to a crash .  both feigenbaum and freund , and sornette et al. also looked at randomly selected time windows in the real data and generally found no evidence of log - periodicity in these windows unless they were looking at a time period in which a crash was imminent . while these results are not tied to any dgp since there is no monte carlo simulation involved , it is still not clear what if any conclusions can be drawn from this result .it is impossible to collect a random sample of any size with independent draws from a single century of data .perhaps in the year 3000 , our descendants will be better able to judge by this method if log - periodicity really is an indicator of financial crashes . for those of us who do not expect to live that long , some more clever approach must be invented if we are to answer this question with any degree of confidence .if we focus on a specific choice of the nonlinear parameters , , , , , and in the specification of eq .( [ logps ] ) , then we have a specification linear in the parameters , , and , and we could naively perform statistical tests on hypotheses regarding the values of these parameters . for example , it would be of great interest to determine if , the coefficient of the term responsible for producing log - periodic oscillations in eq .( [ logps ] ) is significantly different from zero .unfortunately , a minor obstacle gets in the way of our doing this in a consistent manner .our hypothesis is that if , then where is a random disturbance term satisfying =0.\ ] ] there are well - known ordinary least squares ( ols ) estimation formulas both for point estimates of , , and and their corresponding standard errors .these formulas apply if the are homoskedastic and disturbances at different times are uncorrelated . however , if we consider the s&p 500 from january 1980 to september 1987 , the lag-1 autocorrelation and as the lag- autocorrelation . ] for is .9994 , which is not surprising since security prices were once believed to follow random walks .stock prices may move up or down on a given day , but they rarely move far from the previous day s closing .we can eliminate most , if not all , of this serial correlation by looking not at but at the first differences of .define and then the lag-1 autocorrelation for is only .1511 for the same period examined above .let us also define then eq . ( [ simpspec ] ) implies =b\delta f_{1i}+c\delta f_{2i}% \text{.}\label{increment}%\ ] ] we will consider the augmented specification =a+b\deltaf_{1i}+c\delta f_{2i}+d_{2}% \delta_{2i}+d_{3}\delta_{3i}+d_{4}\delta_{4i}.\label{regress}%\ ] ] the additional variables , , and are binary indicator variables ( or so - called `` dummy variables '' ) which take on values of 0 or 1 . for a given , the variable will equal 1 if equals and 0 otherwise .the inclusion of the constant term allows for the possibility of a constant drift in the stock price index .since the magnitude of this drift may depend on the length of time between measurements , we also include the dummy variables .( four is the largest that appears in the data . ) by doing this , we can , for example , correct for weekend effects .these are of concern because weekly cycles in might arise naturally from the institutional details of the market , and , if we did not include them in the regression , an oscillatory function like might easily project onto them , biasing the estimate of its coefficient .[ ptb ]  lpfig8.epsi  in fig .[ lspdiff ] , we plot as a function of time .no obvious oscillatory pattern is apparent in the graph , although this does not prove that no pattern is there .[ c]lll & 1/2/80 - 9/30/87 & 1/2/80 - 6/5/86 + & .000668 & .000450 + & -.006384 & -.019967 + & -.001149 & .001542 + & .001712 & .001646 + & -.002029 & -.002149 + & -.002826 & -.003463 + & 10/7/87 & 6/19/86 + & 0.590 & 0.497 + & 7.333 & 8.255 + & 0.338 & 0.004 + & 13.4 yr & 1285 yr + & 28.80 & -0.79  [ c]lllll & 1958 & & & + & .159777 & & & + & 0.0172 & & & + & & & & + & & std .& & + & & .000273 & 2.434 & .015 + & & .004826 & -1.312 & .190 + & & .000251 & -4.504 & .000 + & & .002153 & 0.791 & .429 + & & .000605 & -3.359 & .001 + & & .001450 & -1.953 & .051  using the same downward simplex method used in section [ logspec ] , we obtain the parameter values listed in the first column of table [ parval2 ] . in table[ estlinu ] , we report the results of the corresponding ols regression , holding the nonlinear parameters fixed ( the linear parameters deviate slightly from table [ parval2 ] because they are computed by a different program ) .in addition to ols standard errors , we report corresponding statistics and nominal values which correspond to the probability that will be greater than or equal to its observed value under a null hypothesis that the given coefficient is zero .note that these nominal values are computed under the assumption of homoskedasticity and no serial correlation .it is fairly well accepted that the s&p 500 is a heteroskedastic process .furthermore , while the autocorrelation of may be insignificant , it is well known that typically shows significant autocorrelation , so the time series is very likely serially dependent . therefore the reported values should not be given much credence .the search for standard error estimators which consistently estimate the standard deviation of regression coefficients , even for a model which exhibits heteroskedasticity and autocorrelation of an unknown nature , is presently an active field of econometric research .we explored both the newey - west ( see chapters 11 - 12 of ) and the kiefer - vogelsang approaches .however , our results were not consistent between these different methods , which suggests that our sample size is too small for the asymptotic probability theory underlying these methods to be a good approximation .consequently , it proved necessary to use monte carlo simulations to estimate the finite - sample probability distribution of the statistics .  in order to perform monte carlo simulations ,the null hypothesis must completely specify the data - generating process ( dgp ) to be simulated .the null hypothesis in this context , which is just the blanket assumption that there is no log - periodicity , is a compound hypothesis and does not fully specify a dgp .however , our main concern is whether any apparent log - periodicty can be accounted for by a simple specification of the index s behavior such as a random walk or a garch process . matching the first and second moments of conditional upon , we constructed a random walk with normally distributed innovations .  in 20 simulations , six produced fits to the specification of ( [ regress ] ) with statistics for larger in magnitude than the observed value of -1.312 , which implies a value of .30 .thus the term is not statistically significant .this is not entirely surprising since the inclusion of the term in eq .( [ simpspec ] ) is really not well motivated in . unlike the term, the term does not come out of sornette and johansen s landau expansion .rather , it is included ad hoc to account for the growth trend seen in most financial indices , though this growth trend could also be accounted for by a constant drift . on the other hand , we observe only one simulation with a statistic for the log - periodic coefficient as large as 4.5 in magnitude .this implies a -value of .050 , so we do not firmly reject the log - periodic term at the 5% level .if the sornette - johansen paradigm has any practical value though , we should be able to predict a crash even if we do not have all the data leading right up to the crash . what happens if we disregard the last year of stock market data ?do the first differences of the s&p 500 contain the information necessary to forecast a crash in october of 1987 ?  [ c]lllll & 1624 & & & + & .128761 & & & + & 0.0143 & & & + & & & & + & & std .& & + & & .000309 & 1.458 & .145 + & & .009796 & -2.062 & .039 + & & .000481 & 3.232 & .001 + & & .002327 & 0.708 & .479 + & & .000678 & -3.169 & .002 + & & .001593 & -2.172 & .030  using a window of data from january 1980 to june 1986 , the best fit we obtain is reported in the second column of table [ parval2 ] , and the corresponding ols regression is given in table [ regress86 ] .notice that we do not forecast a crash in 1987 .rather , we forecast a crash just beyond the end of the window . using the same dgp as before , in five of 20 simulationswe obtained a statistic larger than 3.23 for the log - periodic term , implying .thus the log - periodic term is not significant at the 5% level in this fit .as we noted in section [ logspec ] , the same data set may give several fits with roughly similar values of .although the nonlinear parameters of the fit for the full data set do not give the best fit for the truncated data set , conceivably they could still produce a decent fit with a statistically significant log - periodic component .an ols regression using the nonlinear parameters of the first column of table [ parval2 ] is reported in table [ regress87b ] . in 20 simulations, 14 had statistics for larger in magnitude than 2.767 , so . thus even using the nonlinear parameters that were statistically significant for the full data set , we do not find a statistically significant log - periodic component in the truncated data set .[ c]lllll & 1624 & & & + & .129005 & & & + & 0.0124 & & & + & & & & + & & std .& & + & & .000438 & 0.147 & .883 + & & .016639 & -1.971 & .049 + & & .000318 & -2.767 & .006 + & & .002362 & 0.306 & .759 + & & .000924 & -3.239 & .001 + & & .001829 & -2.288 & .022  since the log - periodic function is significant for the full data set , there must be some structure in the data which projects onto this function. however , the failure of this function to produce a significant coefficient in the truncated data set suggests the pertinent structure is localized within the last year before the crash .note that this structure need not be log - periodic . we can only conclude that the log - periodic function represents this structure better than any of the other functions included in the regression .whatever is happening in the last year , we can conclude that  ] has no log - periodic trend .one might wonder how this can be reconciled with the fits observed in section [ logspec ] .how can behave log - periodically when its expectation does not ?serial correlation may answer this question .a short string of several increments which coincidentally fall in the same direction followed by a string of increments in the opposite direction may create an oscillation completely by happenstance with no underlying force behind it .in their search for an independent form of evidence to corroborate their hypothesis that sizable financial crashes are often the culmination of forces that build up over long periods of time , johansen and sornette examined the frequency distribution of drawdowns , a drawdown being the cumulative drop in a financial index from a local maximum to the succeeding local minimum .[ ddf ] displays a semilog graph of the frequency distribution of drawdowns larger than 0.5% for the s&p 500 during the period 1962 - 1998 .if we disregard the lone drawdown of magnitude larger than 20% , which corresponds to the october 1987 crash , it is immediately apparent that this semilog graph looks linear . seizing upon this observation , sornette andjohansen concluded that there must be two populations of drawdowns .the majority of drawdowns have an exponential distribution , and these constitute the first population .those drawdowns , like the 87 crash , which do not obey this exponential distribution constitute the second population .sornette and johansen further suggest that the first population is produced by the normal behavior of a financial index obeying a garch process .in contrast , the second population is the result of the imitative behavior of investors , and it is this population that exhibits log - periodic oscillations prior to the drawdown .[ ptb ]  while the frequency distribution of the first population of drawdowns ( every data point except the 87 outlier ) may look exponential , we can objectively test whether it actually is exponential . in table[ ddr ] , we present the results of a regression of the logarithm of the frequency versus a quadratic function of the drawdown size .the alleged outlier , the october 87 crash , was not included as a data point in this regression . nevertheless , the quadratic term in this regression with the 87 crash excluded is statistically significant at the 1% level .the justification for treating the 87 crash separately was that it did not fit to a line through the other drawdowns .however , the significance of the quadratic term indicates the logarithm of the frequency distribution for the remaining drawdowns is not linear either . since a nonexponential frequency distribution can certainly be found that fits to all the data , the visual evidence of two populations is unconvincing .[ c]ccccc # of obs .& 12 & & & + & .180 & & & + & .969 & & & + & & & & + & & standard error & & + & & 0.155 & -7.065 & 0.000 + & & 0.012 & 3.378 & 0.008 + constant & & 0.439 & 17.801 & 0.000  aware of these deviations from the exponential curve , johansen and sornette have recently proposed ( ) that the first population is better described by a `` stretched '' exponential where is the frequency of drawdowns of size .they find that .using this specification on our data set , we find the best value of within the interval  ] must also be rejected .[ c]ccccc # of obs .& 12 & & & + & .196 & & & + & .9666 & & & + & & & & + & & standard error & & + & & 0.230 & -6.722 & 0.000 + & & 0.010 & 2.280 & 0.049 + constant & & 0.534 & 15.640 & 0.000  however , if we allow to range beyond this interval , we find that gives the best fit , and if we include a term linear or quadratic in neither is statistically significant at the 5% level .furthermore , if we consider a specification of the form then we are able to accomodate all the data including the 87 outlier . in table[ ddr3 ] , we give the regression estimates for such a specification and we obtain a favorable , so with this data set it does not seem reasonable at all to treat the 87 crash differently from other drawdowns . nevertheless , it is possible that this result is an artifact of the small data set available to us .johansen and sornette considered the specification ( [ stretched ] ) over the distribution of drawdowns for a much larger time period .they also looked at many other financial time series besides the s&p 500 , and they consistently obtained values of . in more than half of these examples ,though , it is patently obvious that the specification ( [ stretched ] ) is wrong , for statistically significant deviations are clearly visible in their graphs . if the model is misspecified , no valid inferences can be drawn from it .[ c]ccccc # of obs .& 13 & & & + & .289 & & & + & .9519 & & & + & & & & + & & standard error & & + & & 0.570 & -9.023 & 0.000 + & & 0.089 & 6.069 & 0.000 + constant & & 0.842 & 14.071 & 0.000  of course , in section [ logspec ] , we saw that the 87 crash was not the only drawdown between 1962 and 1998 with a log - periodic precursor .if we had an objective way of distinguishing drawdowns with log - periodic precursors from drawdowns without such precursors , then it is still possible that the distribution of drawdowns without precursors is exponential .while the evidence of log - periodic precursors to financial crashes has spawned a minor industry in the physics community , it is apparent that a more rigorous investigation of this phenomenon must be conducted before we can come to a firm conclusion that log - periodic oscillations reflect a discrete scale invariance underlying the economy and furthermore that such oscillations signal an impending crash .in particular , it would be desirable for researchers to agree upon an objective definition of what constitutes a log - periodic spell so that a statistical association between these spells and ensuing crashes can be firmly established .we must emphasize that a physicist who endeavors to successfully carry out these goals will have to employ and perhaps even invent advanced statistical methods of a nature that may be unfamiliar to him because they are simply unnecessary in more customary physics settings .a physicist can typically rely more confidently on elementary statistical methods because he can do controlled experiments .he can often identify and directly measure the relevant sources of error .testing a model can be just a matter of determining whether the model agrees with the data to within a degree that can be accounted for by these known sources of error .in contrast , an econometrician s task is vastly more difficult , at least as far as testing models goes , for he must simultaneously estimate the parameters of his model and the properties of any unmodeled disturbances using the same data set ( which was probably gathered under less than ideal conditions ) . as a result, econometricians have had to acquire complicated tools to efficiently salvage as much information as they can from their data , and physicists who expand into this realm will have to do the same .while the present work does not refute what has come before , we think it does shed some doubt on the hypothesis of discrete scale invariance in financial markets . though log - periodicity is visually apparent in the s&p 500 preceeding the 87 crash, we find no evidence of log - periodicity in the mean function of this index prior to a year before the crash .this would suggest that if log - periodicity is embedded in the stochastic process of the index , then it must be conveyed in higher - order moments. however , evidence of log - periodicity in the second moment of the process is also lacking . in fig .[ volatility ] , we plot a measure of the volatility of the s&p 500 , the standard deviation of sampled over a moving window of fifty trading days , from january 1980 to october 1987 .no pattern of log - periodicity can be seen in the graph .the negative result of this paper is also troubling from a theoretical perspective . while johansen et al .( ) are not alone in their efforts to construct an economic model which can produce log - periodic price series ( see for example ) , economists would be reluctant to embrace any theory that diverges from johansen et al.s attempt to ensure the model is robust to the introduction of a rational agent with rational expectations .there are today economic theorists exploring models in which agents are not fully rational , but a model whose predictions absolutely depend on there being no one in the economy who has a good grasp of how the world works is not likely to acquire a large following .any model which does not suffer from this disease will have to satisfy a restriction like the martingale condition of eq .( [ gen mar2 ] ) .a model which incorporates discrete scale invariance and log - periodicity in its structure would not necessarily imply that the expectation of prices must also behave log - periodically .however , if log - periodicity arises because the probability of a crash behaves log - periodically , then it is difficult to imagine how one could avoid making the prediction that the expected price must also behave log - periodically .if nothing else , the results of this paper would seem to demonstrate that a more circuitous explanation of log - periodicity is required .

Abstract: motivated by the hypothesis that financial crashes are macroscopic examples of critical phenomena associated with a discrete scaling symmetry , we reconsider the evidence of log - periodic precursors to financial crashes and test the prediction that log - periodic oscillations in a financial index are embedded in the mean function of this index . in particular , we examine the first differences of the logarithm of the s&p 500 prior to the october 87 crash and find the log - periodic component of this time series is not statistically significant if we exclude the last year of data before the crash .  we also examine the claim that two separate mechanisms are needed to explain the frequency distribution of draw downs in the s&p 500 and find the evidence supporting this claim to be unconvincing .