a complex system is inherently sensitive to failures of its components .one must therefore determine maintenance policies in order to maintain an acceptable operating condition .optimizing the maintenance is a very important problem in the analysis of complex systems .it determines when it is best that maintenance tasks should be performed on the system in order to optimize a cost function : either maximize a performance function or conversely minimize a loss function .moreover , this optimization must take into account the random nature of failures and random evolution and dynamics of the system .the example considered here is the maintenance of the heated hold - up tank , a well know test case for dynamic reliability , see e.g. .the system consists of a tank containing a fluid whose level is controlled by three components : two inlet pumps and one outlet valve .a thermal power source heats up the fluid .the failure rate of the components depends on the temperature , the position of the three components monitors the liquid level in the tank , and in turn , the liquid level determines the temperature .the main characteristic of this system is that it can be modeled by a stochastic hybrid process , where the discrete and continuous parts interact in a closed loop . as a consequence , simulating this process and computing related reliability indices has been a challenge for the dynamic reliability community . to our best knowledge , optimization of maintenance policies for the heated hold - up tank has not been addressed yet in the literature .the only maintenance operation considered here is the complete replacement of all the failed components and the system restarts in its initial equilibrium state .partial repairs are not allowed .mathematically , this problem of preventive maintenance corresponds to a stochastic optimal stopping problem as explained by example in the book of aven and jensen .it is a difficult problem because of the closed loop interactions between the state of the components and the liquid level and temperature .a classical approach consists in using condition - based maintenance ( cbm ) to act on the system based on its current state and before its failure .one can for example calculate the remaining useful life ( rul ) of the system and the preventive replacement is carried out when the deterioration level exceeds a certain threshold or enters in a certain state .our approach also takes into account the current state of the process , but our decision rule is not based on damage accumulation nor does it correspond to hitting some threshold .instead , it involves a performance function that reflects that the longer the system is in a functioning state the better .the dynamics of the heated hold - up tank can be modeled by a piecewise deterministic markov process ( pdmp ) , see .therefore , our maintenance problem boils down to an optimal stopping problem for pdmp s .pdmp s are a class of stochastic hybrid processes that has been introduced by davis in the 80 s .these processes have two components : a euclidean component that represents the physical system ( e.g. temperature , pressure , ) and a discrete component that describes its regime of operation and/or its environment .starting from a state and mode at the initial time , the process follows a deterministic trajectory given by the laws of physics until a jump time that can be either random ( e.g. it corresponds to a component failure or a change of environment ) or deterministic ( when a magnitude reaches a certain physical threshold , for example the pressure reaches a critical value that triggers a valve ) .the process restarts from a new state and a new mode of operation , and so on .this defines a markov process .such processes can naturally take into account the dynamic and uncertain aspects of the evolution of the system .a subclass of these processes has been introduced by devooght for an application in the nuclear field .the general model has been introduced in dynamic reliability by dutuit and dufour .the objective and originality of this paper is twofold .first , we propose an optimization procedure for a well - known benchmark in the dynamic reliability literature .the tank model was first introduced by where only one continuous variable ( liquid level ) is taken into account , and then in and where the second variable ( temperature ) is introduced .they have tested various monte carlo approaches to simulate the process to compute reliability and safety indices . in ,the authors have used the same system to present continuous cell - to - cell mapping markovian approach ( cccmt ) still to simulate the process .the simulation of the holdup tank example has been and is still widely studied in the literature ( not exhaustive ) . herewe go one step further and not only propose to simulate the tank process but also we optimize it .second , even though pdmp s have been recognized as a powerful modeling tool for dynamic reliability problems , there are very few numerical tools adapted to these processes .our aim is to further demonstrate the high practical power of the theoretical methodology described in , by applying it to the tank benchmark . in ,the authors have proposed a numerical algorithm to optimize pdmp s and have studied its theoretical properties .this optimization procedure was first applied to an example of maintenance of a metallic structure subject to corrosion , without closed loop interactions or deterministic jumps .in addition , the system has only one continuous variable and the cost function is simple and does not depend on time , see . in this paper , we adapt the numerical procedure proposed in to the more challenging heated hold - up tank problem with two continuous variables , deterministic jumps when these variables hit some given boundaries and closed loop interactions between continuous and discrete variables .furthermore , we consider a cost function that depends on both continuous variables as well as on the running time .the remainder of this paper is organized as follows . in section [ section :model ] , the dynamics of the heated hold - up tank is presented with more details as well as the framework of pdmp s . in section [ section : opti ] the formulation of the optimal stopping problem for pdmp s and its theoretical solution are briefly recalled and the four main steps of the algorithm are detailed . in section [ section: res ] the numerical results obtained on the example of the tank are presented and discussed . finally , in section [ section : conclusion ] a conclusion and perspectives are presented .we are interested in the maintenance of a heated hold - up tank .the dynamics of the tank can be modeled by a piecewise deterministic markov process ( pdmp ) .we first describe with more details the dynamics of the tank , then we recall the definition and some basic properties of pdmp s .the tank model is a well known benchmark in dynamic reliability .it was first introduced by where only one continuous variable ( liquid level ) is taken into account , and then in and where the second variable ( temperature ) is introduced .we have kept the values of the parameters defined in those papers .the system is represented on figure [ fig1 ] .it consists of a tank containing a fluid whose level is controlled by three components : two inlet pumps ( units 1 and 2 ) and one outlet valve ( unit 3 ) .a thermal power source heats up the fluid .the variables of interest are the liquid level , the liquid temperature and the state of the three components and the controller .each component has four states : on , off , stuck on or stuck off . once a unit is stuck ( either on or off ) it can not change state .the possible transitions between these four states are given in figure [ fig2 ] .thus , by a random transition a working unit can only become stuck ( either on or off ) .the initial state of the components is on for units 1 and 3 and off for unit 2 .  ,width=226 ]  the intensity of jumps for unit depends on the temperature through the equation with given in eq .( [ eq : a ] ) , see   function is represented on figure [ fig3 ] and the various parameters come from the literature , see , and are given in table [ tab1 ] .the special form of the failure rate as a product of a constant depending on and a function of the temperature allows for all three units to have failure rates with the same dependence on the temperature , but different scaling parameters .indeed , at the reference temperature of , the mean time to failure of unit is , for unit , it is and for unit it is .in addition , the shape for function was chosen in the original benchmark so that there is a very high failure rate when the temperature is high .more specifically , the parameters are chosen such that is lowest ( equal to ) when the temperature is equal to a reference temperature of , it equals when the temperature is and it is highest ( equal to ) when the temperature equals the critical temperature of .the exponential functions are chosen in order to enable this very high dependence with the temperature .roughly speaking , the units fail times more often when the temperate is than when it is .   as a function of ,width=188 ]  in addition, control laws are used to modify the state of the components to keep the liquid within two acceptable limits : meters and meters .if the liquid level drops under , the components 1 , 2 , 3 are put respectively in state on , on and off ( provided they are not stuck ) .if the liquid level rises above , the components are put respectively in the state off , off and on ( provided they are not stuck ) . unlike the classical model presented in , we also allow the control unit to fail . at each solicitation, the control may succeed with probability independently from previous successes .once it has failed , it will never succeed again .therefore the control unit has two possible states : working or failed ..parameters for the tank dynamics [ cols="^,^",options="header " , ]the numerical method described in has been applied to a well known test case of dynamic reliability to approximate the value function of the optimal stopping problem and an -optimal stopping time for a piecewise - deterministic markov process , that is the maintenance date for the tank .the quantization method proposed can sometimes be costly in computing time , but has a very interesting property : it can be calculated off - line .moreover it depends only on the dynamics of the model , and not on the cost function chosen , or the actual trajectory of the specific process one wants to monitor .the calculation of the optimal maintenance time is done in real time , making our procedure applicable in practice .the optimal maintenance time is updated at the changes of mode and has a conditional threshold form , which allows scheduling maintenance services in advance .if one only changes the reward function without changing the dynamics of the tank , one just has to run the optimization part of the algorithm , and not the quantization grids .this can be done in real time . if one wants to change the dynamics of the system , or add some components , one has to rewrite the simulation code for the system , and with this new code re - run the quantization grids , which can be quite long .however , the general methodology is valid for a wide class of piecewise deterministic markov processes and not at all specific to the tank .the method has been implemented on the heated hold - up tank .the main characteristic of this system is that it can be modeled by a stochastic hybrid process , where the discrete and continuous parts interact in a closed loop .the optimization problem under study has no analytic solution .however , our method is based on a rigorous mathematical construction with proof of convergence .in addition , simple comparisons between no motoring and our policy also prove its practical validity with a significant improvement of the performance of the system ( the mean performance is increased by and the top events are almost always avoided ) .our next project is to extend this research in two main directions .first , we could allow only partial repair of the system .the problem will then be to find simultaneously the optimal times of maintenance and optimal repair levels .mathematically , it is an impulse control problem , which complexity exceeds widely that of the optimal stopping .second , our method requires a perfect observation of the state process at the jump times .it would be interesting to extend our results to a noisy observation of the process , as often happens in real life .this work was partially funded by the french national research , anr grant fautocoes , program arpege , anr-09-segi-004 .the authors also thank pr .franois dufour for initial discussions on how to define an interesting optimal stopping problem for the test - case of the tank .this paper was presented at esrel 2012 conference .

Abstract: we present a numerical method to compute an optimal maintenance date for the test case of the heated hold - up tank . the system consists of a tank containing a fluid whose level is controlled by three components : two inlet pumps and one outlet valve . a thermal power source heats up the fluid . the failure rates of the components depends on the temperature , the position of the three components monitors the liquid level in the tank and the liquid level determines the temperature . therefore , this system can be modeled by a hybrid process where the discrete ( components ) and continuous ( level , temperature ) parts interact in a closed loop . we model the system by a piecewise deterministic markov process , propose and implement a numerical method to compute the optimal maintenance date to repair the components before the total failure of the system .


silicon photomultipliers ( sipm ) have excellent time and photon counting resolutions , among other qualities .however , crosstalk and afterpulsing seriously limit the performance of these photon detectors .these phenomena are the production of parasitic avalanches in either the pixel where a primary avalanche was triggered ( afterpulsing ) or a neighboring one ( crosstalk ) .their main effects are to increase both the count rate and the total charge collected for an input light pulse , e.g. , from a scintillator .the physical processes considered in our analysis are illustrated in figure [ fig : processes ] . in the first place , a charge carrier of a primary avalanchecan be trapped in a crystal defect within the pixel , then when the avalanche has already been quenched , the carrier may be released triggering a new avalanche .we refer to this process as afterpulsing due to traps ( ap - traps ) and it has as many components as types of traps in the silicon lattice . in the second place , secondary photons of the primary avalanche may reach neighboring pixels triggering almost simultaneously other avalanches .this is the so - called optical crosstalk ( ct - opt ) .some sipms incorporate trenches filled with an optical absorber surrounding pixels to prevent this effect , but ct - opt can still be produced by photons reflected on the bottom surface of the detector .finally , minority carriers generated in the silicon substrate by secondary photons of the primary avalanche can diffuse back to either the primary pixel or to a neighboring one , contributing respectively to crosstalk ( ct - diff ) or afterpulsing ( ap - diff ) .the ct - diff is usually referred to as delayed crosstalk because of the relatively long diffusion times , in contrast to the ct - opt , which is also called prompt crosstalk .we presented in an experimental method and an analytical model to describe the ct - opt in a sipm fabricated by hamamatsu . in the present work, our study has been extended with new measurements for the same sipm as well as for a second sipm , also from hamamatsu .in addition , we have constructed a statistical model accounting for ap - traps , ct - diff and ap - diff , and preliminary measurements to validate it are shown .a similar analysis of afterpulsing and ct - diff in a stmicroelectronics sipm has recently been presented in .the experimental method has been described in detail in , and only an overview is given here .the setup consists of the sipm with the associated bias circuit , a fast amplifier , and a digital oscilloscope to register and store the signals for later analysis .two sipms from hamamatsu were tested : the s10362 - 11 - 100c ( the one previously used in ) and the s10362 - 33 - 100c .both detectors have a pixel pitch of 100  m and basically the same characteristics , but the first sipm has 100 pixels while the second one has 900 pixels .measurements were performed at dark conditions , that is , an only primary pixel is expected to be triggered at a time .so far , we have characterized the intrinsic crosstalk and afterpulsing ( i.e. , no scintillator was attached to the detector ) at room temperature .an algorithm has been developed to perform a detailed waveform analysis of the recorded signals .this dedicated software calculates numerically the deconvolution of the signal with a negative exponential function that describes the decay of pulses ( time constant of 23 ns for both detectors ) . this way ,pulses are identified as distinct peaks in the deconvolved signal even if they overlap in the original signal .the position and height of the deconvolution peak provide us relative measures of the arrival time and amplitude of each pulse .the algorithm resolves and measures the parameters of pulses as close as 6 ns .in addition , events with two or more pulses with smaller time differences ( down to  ns ) are identified by analyzing the shape and width at threshold level of the deconvolved signal .this allowed us to characterize separately the nearly instantaneous ct - opt and the other phenomena that result in delayed secondary pulses .a more precise measurement of the pulse amplitude is obtained as the pulse height with baseline substraction . to do that, the software performs an exponential extrapolation of the original signal in a small region prior to each pulse .the ct - opt gives rise to pulses with amplitudes two or more times higher than that corresponding to only one triggered pixel .the relative intensities of the pulse amplitude spectrum measured at dark conditions thus provides the probability distribution of the total number of simultaneously triggered pixels per primary avalanche , i.e. , ct - opt excitations .we produced this spectrum using the pulse height with baseline substraction to optimize the resolution in .in addition , quality cuts were applied to reject low - amplitude pulses due to pixels that were triggered while being recharged after the last avalanche breakdown .we presented in a statistical model based on the assumption that secondary photons can only reach a certain neighborhood of pixels around the primary one .cascades of ct - opt excitations propagating through the whole array of pixels were included , but taking into account that pixels that have already been triggered can not be triggered twice .the distribution was then calculated by identifying all the ct - opt `` histories '' that contribute to each probability considering four possible arrangements of neighbors : the 4 nearest neighbors , the 8 nearest neighbors , the 8 l - connected neighbors and all neighbors ( see for details ) . for simplicity, we assumed the same ct - opt probability for any neighbor and ignored the boundaries of the array of pixels for the calculations .analytical expressions for the mean and variance of the distribution were also obtained .  in figure[ fig : crosstalk ] , the model predictions for the four arrangements of neighbors are compared with the experimental probabilities obtained at a given bias voltage for both the s10362 - 11 - 100c detector ( top plot , taken from ) and the s10362 - 33 - 100c detector ( bottom plot ) . in particular , the probability ratio ( theoretical over experimental ) is represented versus from 1 to 5 , where the model predictions were normalized so that the first probability matches the experimental one , i.e. , their ratio is unity .the error bars of the remaining ratios were determined propagating the uncertainties of the experimental probabilities .the comparison includes two previous analytical models described in , where is assumed to follow a geometric or a borel distribution .as explained in , these two models can be regarded as limit situations of ours .   for optical crosstalk over the experimental ones measured at dark conditions and a given bias voltage .results for the two tested detectors are consistent with the hypothesis of the 4 nearest neighbors. see text for details.,title="fig : " ]  for optical crosstalk over the experimental ones measured at dark conditions and a given bias voltage .results for the two tested detectors are consistent with the hypothesis of the 4 nearest neighbors .see text for details.,title="fig : " ]   distribution for the s10362 - 11 - 100c detector .the legend shows the order from top to down of the lines representing the model predictions.,title="fig : " ]  distribution for the s10362 - 11 - 100c detector .the legend shows the order from top to down of the lines representing the model predictions.,title="fig : " ]  in figure [ fig : crosstalk_v ] , the model predictions of the mean and variance of the distribution are compared with new experimental results for the s10362 - 11 - 100c detector as a function of the overall ct - opt probability , defined as .the overvoltage , i.e. , bias voltage minus breakdown voltage , at which measurements were performed is also indicated for each data point ( the overall ct - opt probability grows quadratically with overvoltage ) .  the comparisons shown in figures [ fig : crosstalk ] and [ fig : crosstalk_v ] reveal that the only model consistent with our experimental data is the one assuming that ct - opt solely takes place in the 4 nearest neighbors of each pixel .this is the case for both tested detectors , which have different array sizes .therefore , it can also be concluded that border effects are unimportant to describe ct - opt effects at these conditions .at dark conditions , the distribution of the pulse arrival time measured with respect to the previous pulse is a combination of the contributions of both uncorrelated dark noise and correlated secondary pulses ( i.e. , ap - traps , ap - diff and ct - diff ) .since we were interested in measuring the probability of secondary pulses per primary avalanche , we obtained this time distribution selecting pairs of pulses in which the first one ( the primary ) has an amplitude corresponding to one only triggered pixel , that is , without ct - opt .in addition , this first pulse of the pair was required to be at least 500 ns far from previous pulses to avoid mixing contributions of secondary pulses from different primaries .assuming that the total number of secondary pulses of each type per primary avalanche is poisson distributed , the above time distribution can be expressed as   \nonumber\\ & \left[r_{\rm dc}+\sum_i\lambda_i\cdot f_i(t)\right]\cdot{\rm d}t\ , , \label{eq : afterpulsing}\end{aligned}\ ] ]  where  ns is the minimum time difference allowed between pulses due to analysis limitations , is the dark count rate , is the mean number of secondary pulses of type per primary avalanche , and is the corresponding normalized time distribution .the details of this model , in particular the parameterization of the functions , are still under progress and will be explained in a paper in preparation .only the general aspects are given below .the distribution for ap - traps was characterized by the product of three factors : a negative exponential distribution describing the release of trapped carriers ; the exponential pixel recharging after the primary avalanche , which is proportional to both the gain and the probability of the secondary avalanche ; and the -dependent fraction of afterpulses ( of any type ) with amplitudes above the detection threshold applied in the analysis software .the last two factors , which are also applicable to the distribution for ap - diff , account for the steep break of the experimental time distribution between 20 and 30 ns ( see figure [ fig : afterpulsing ] ) and were determined from the time - amplitude information of afterpulses with short delay .a monte carlo simulation was carried out to describe the timing and relative contributions of the ap - diff and ct - diff effects ( bottom plot of figure [ fig : processes ] ) .values for the physical parameters related to the emission and absorption of secondary photons , the diffusion of minority carriers and the layer configuration of the chip were varied within a wide interval around typical data taken from the literature ( e.g. , see ) .preliminary simulation results showed that the transient time distribution can be properly described by the product of the exponential decay of the number of minority carriers due to recombination mechanisms and a negative power law .also , the relative contributions of ap - diff and ct - diff were found to be strongly dependent on the pixel pitch .for instance , for a very small pixel pitch of 10  m , the ap - diff component is negligible compared to the ct - diff one , while both components are significant for a pixel pitch of 100  m ( present case ) .a fit of our model to experimental data for the s10362 - 11 - 100c detector at an overvoltage of  v is shown in figure [ fig : afterpulsing ] , where the contributions of the different types of secondary pulses and of dark counts are represented by thin lines .the corresponding accumulated contributions are also shown in the inset of the figure in a linear scale .this result corresponds to a given set of parameters assumed in the above simulation , which involves that the distributions and the relative contributions of ap - diff and ct - diff are fixed ( i.e. , only one parameter for both components is let free in the fit ) . in this case ,two ap - traps components with different mean release time were necessary to describe the data : a fast component with  ns and a slow one with  ns .on the other hand , when letting all the parameters related to ap - diff and ct - diff free , good quality fits were also obtained even omitting any ap - traps component .similar results were obtained for the s10362 - 33 - 100c detector ( not shown here ) .we are making the simulation more realistic in order to be more restrictive on the fitting parameters .ns are also shown in the inset ( linear scale ) . ]  even though the number of ap components has not been determined yet , the sum of the parameters of all the ap components was found to be nearly independent of the assumptions made on the fit .the parameter associated to the ct - diff was roughly fit independent as well , because it is basically determined by data at  ns , where ap contributions can be neglected .these parameters , i.e. , the one for the total contribution of afterpulsing and the one for ct - diff , are represented in figure [ fig : afterpulsing_v ] as a function of overvoltage , showing a pure quadratic behavior , as expected .parameters , but using the distributions determined in the fit shown in figure [ fig : afterpulsing ] at a given bias voltage .appropriate corrections were applied to account for the overvoltage dependence of the detection threshold effects on the afterpulsing measurements . ]  .the uncertainties are statistical only . ]we have developed an experimental method based on a waveform analysis to characterize the crosstalk and afterpulsing in sipms .results were obtained for the s10362 - 11 - 100c and s10362 - 33 - 100c detectors from hamamatsu at dark conditions and room temperature .measurements of optical crosstalk have been compared with an analytical model that we presented in a previous publication .our data for both detectors are consistent with the hypothesis that optical crosstalk is only possible between adjacent pixels .  a statistical model of afterpulsing anddelayed crosstalk has been constructed including a detailed parameterization of the time distributions of the different types of secondary pulses .our preliminary results show a significant probability of afterpulsing in both detectors , although the characterization of its components is still under study .the delayed crosstalk was also found to be important , mostly at short time .these studies are being extended to characterize other sipms and for a larger variety of conditions .we intend to apply our models to describe crosstalk and afterpulsing effects on particular experimental cases .this work was supported by mineco ( fpa2012 - 39489-c04 - 02 ) and consolider cpan csd2007 - 42 .

Abstract: an experimental method to characterize the crosstalk and afterpulsing in silicon photomultipliers has been developed and applied to two detectors fabricated by hamamatsu . an analytical model of optical crosstalk that we presented in a previous publication has been compared with new measurements , confirming our results . progresses on a statistical model to describe afterpulsing and delayed crosstalk are also shown and compared with preliminary experimental data .   silicon photomultipliers , crosstalk , afterpulsing


in many soft - matter and biological systems electrostatic interactions have a strong influence on the physical behavior . a typical example are charge stabilized colloidal dispersions , whose structure is mainly determined by the interplay between van der waals attraction and electrostatic repulsion .a simplifying feature is the fact that it is often sufficient to describe the structure of the `` cloud '' of surrounding ions ( counterions and salt ions ) just in terms of mean field theory , in particular if the ions are monovalent . at the center of this theory is the well known poisson boltzmann equation , which , from the mathematical point of view , is a nonlinear partial differential equation .analytical work in this field has been mainly confined to the linearized ( debye hckel ) version , which however is often insufficient , in particular near strongly charged objects .this has prompted efforts to develop methods to solve the equation numerically , ideally in three dimensions and without restrictions on the underlying geometry or spatial symmetry .these are usually based on standard finite difference or finite element techniques ( for a recent review see ref . ) , and have meanwhile reached a substantial degree of sophistication and complexity .recently , however , maggs has put forward a completely new approach to electrostatics in soft matter research , which bears a certain similarity to lattice gauge theories .the central idea is to use the _ electric field _ instead of the electrostatic potential as the quantity on which the algorithm operates , and to view gauss law as a _ constraint _ for the field configurations . on a lattice , it is then easy to construct an electric field ( ) configuration which satisfies gauss law for a given ( arbitrary ) charge density .the hard part is rather the transversal part of the field , which should satisfy , but initially does not ( unless one uses a sophisticated initialization procedure based upon solving the poisson equation ) .this transversal degree of freedom can then be removed by local relaxations ( this is the approach taken in the present paper ) , or integrated out by performing a monte carlo or molecular dynamics simulation on the overall system .a crucial aspect of the method is to locally update both and simultaneously in a way that gauss law is still satisfied _ after _ the update , i. e. the `` constraint surface '' is never left .  the original maggs approach is based upon a system of discrete charges whose statistical physics is treated in a consistent way .however , it is also possible to apply this to the mean field version of the theory , i. e. the poisson boltzmann equation .the purpose of the present paper is to outline how this can be done in practice . in sec .[ sec : derivation ] we derive the method by re formulating the poisson boltzmann theory in terms of a free energy functional , which is minimized by using a maggs type algorithm .in contrast to previous formulations , this functional provides a true minimum , and therefore the procedure is very straightforward and simple , unconditionally stable , and has a rather modest storage requirement which scales only linearly with the number of grid points . in terms of computational speed , the method can probably not yet compete with the existing packages ; however , it is reasonable to assume that more advanced versions that combine the basic methodology with acceleration techniques like adaptive mesh refinement and / or unstructured meshes , may become a very useful tool .section [ sec : numerics ] presents numerical results obtained with our current simple implementation , while sec .[ sec : speedup ] discusses optimizations based upon fast fourier transforms and hierarchical pre conditioners .finally , sec . [sec : conclus ] finishes with some concluding remarks .consider a system of fixed charges , with total charge ( denotes the elementary charge ) dispersed in a solvent with dielectric constant .the system is confined to a three dimensional finite domain of volume .the charges can be distributed either in macroscopic particles or in any other kind of boundary .furthermore , the domain contains counterions of total charge such that the system as a whole is charge neutral .the total number of counterions is , such that , where is their valence .furthermore , the presence of other ionic species ( salt ions ) is allowed if they satisfy charge neutrality , i. e. if , where is the valence of the ionic species and is its number .the continuum theory provides equations for the concentrations ( particle number densities ) of each ionic species and the electrostatic potential . in the stationary regime ( no time dependence ) and in thermal equilibrium characterized by the thermal energy , they take the form  the first equation is the equilibrium version of the nernst planck equation , which balances the diffusion current of ionic species against the drift caused by the electric field .the second equation is the poisson equation , taking into account all ionic species as a source term , while the fixed charges appear as boundary conditions .the total number of particles is obtained by integrating the concentration over the whole domain : and the charge neutrality may then be expressed by with by integrating eq .[ eq : nernst_planck ] one obtains where the integration constant has the dimension of a concentration , and , for normalization reasons , must have the value inserting this result into eq .[ eq : poisson ] , one obtains the poisson boltzmann equation : which is an equation for only .however , for the algorithm to be discussed below , it will be advantageous to rather consider the equivalent coupled set of equations ( eqs .[ eq : nernst_planck ] , [ eq : poisson ] ) .the poisson boltzmann equations can be rewritten in terms of nondimensional quantities , i. e. in reduced units .the argument of the exponential in eq .[ eq : poisson_boltzmann ] suggests the most natural way of rescaling the potential : i. e. the reduced potential is the electrostatic energy of an elementary charge in units of the thermal energy .this leads to where is the bjerrum length . introducing a parameter as a characteristic length scale ( see below ) ,the gradient operator is rescaled via .this allows writing the equations in the nondimensional form where is the reduced concentration . in terms of the electric field , the equations are the normalization condition for the amount of species is transformed to with the choice of the parameter is completely immaterial for the mathematical formulation of the problem .it is only important to map the numerical results back onto a physical system , and therefore a matter of convention . for many applications ,the choice turns out to be quite useful : this is the natural screening parameter in the finite volume version of linearized poisson boltzmann theory ( therefore it also appears in the corresponding treatment of asymmetric electrolytes ) . in the case of only one monovalent ionic species ( the counterions ), this reduces to it should be noted that in this case .  from now on , we will be concerned with the problem of numerically solving the reduced set eqs .[ eq : nernstplanck3][eq : curl3 ] . inwhat follows , the primes will be omitted , with the understanding that all quantities ( including and ) are given in reduced units .   following the ideas of maggs and rosetto , the poisson boltzmann equation can be re formulated as a constrained variational problem , where a free energy functional is minimized .this functional is constructed such that its euler lagrange equations are equivalent to eqs .[ eq : nernstplanck3][eq : curl3 ] .its form is the first term corresponds to the electrostatic energy and the second to the entropy .gauss law and the mass normalization conditions are included as constraints , via lagrange multipliers : the field is the electrostatic potential , while the numbers are the chemical potentials of the species .the euler lagrange equations for this variational problem will be derived next .variation with respect to and just recovers the constraint equations variation with respect to results in while variation with respect to yields the dependence on the unknown lagrange multipliers is removed by taking the gradient of eq .[ eq : variation1 ] and the curl of eq .[ eq : variation2 ] to obtain finally , inserting eq .[ eq : variation2 ] into eq .[ eq : variation3 ] leads to in summary , the derived equations ( eqs .[ eq : variation5 ] , [ eq : constraint1 ] , [ eq : variation4 ] ) are the desired set in other words , the solution of the minimization problem is identical to the solution of the poisson boltzmann equation .  in this context, one should make a few important observations .firstly , we note that solving the euler lagrange equations will provide one and _ only one _ solution : it has been proven ( by variational methods ) that the poisson boltzmann equation has one unique solution .secondly , it is easy to show that the solution is indeed a local minimum : by writing where , form the solution of the problem , and , are small deviations which satisfy the constraints , i. e. one finds i. e. small deviations from the solution will always increase the functional .together with the uniqueness , this shows that the functional has one and only one minimum , which corresponds to the solution of the poisson boltzmann problem .therefore , a procedure that relaxes all degrees of freedom of the functional such that it is systematically decreased , while staying on the constraint surface , will ultimately run into the one and only minimum of the free energy landscape .the iterative procedure may be slow and hampered by small eigenvalues of the hessian at the minimum , but such problems can be kept under control by careful convergence checks and variation of the number of iterations .a simple algorithm that initializes the system on the constraint surface , keeps it there , and systematically decreases by local updates of all non constrained degrees of freedom will be outlined in sec .[ sub : algorithm ] . since the constraints are always satisfied during the procedure , the lagrange multiplier terms in the functional may be omitted , such that it is simplified to it should also be noted that previous approaches that were also based upon a free energy functional ( see , e. g. , ref . ) , did _ not _ truly search for a minimum , but rather for a _ saddle point_.these methods were not based upon a functional that involves the electric field , but rather one that depends on the electrostatic potential , where the are , as before , the lagrange multipliers corresponding to the mass normalization conditions . however , in contrast to its meaning in eqs .[ eq : define_functional_1 ] , [ eq : define_functional_2 ] , is here _ not _ a lagrange multiplier , but rather a degree of freedom .it is straightforward to show that the euler lagrange equations of this problem are equivalent to the poisson boltzmann equation . to show that this solution is indeed not a minimum but rather a saddle point , one again decomposes and into the solution plus small deviations , with this yields i. e.the quadratic form of the deviations is _ not _ positive definite .it is natural to suspect that this lack of positive definiteness causes various numerical difficulties in terms of stability , which therefore are intrinsically absent in the new formulation .the computational domain is a rectangular parallelepiped of size with periodic boundary conditions .this box is discretized by a simple orthorombic ( usually : cubic ) lattice with sites and lattice spacings , enumerating the cartesian directions . the volume of a unit cell is thus .the concentrations are variables on the sites , while the electric field is associated with the _ links_. the positions of the concentration fields are the vectors where are integers .the field is located at the positions similarly , the positions for and are respectively .furthermore , it is useful to define these definitions allow to approximate the functional by and to also discretize the divergence operator in a straightforward way : gauss law then reads introducing fluxes via this is rewritten as finally , the normalization condition for the amount of ionic species is discretized as    the numerical minimization procedure starts from some configuration of the discretized fields and which satisfies all the constraints , i. e. the normalization conditions for the ions , plus gauss law .the algorithm then performs successive local changes in the electric fields and the concentrations , analogously to the monte carlo moves of maggs and rosetto .these moves have the big advantage that they rigorously conserve the constraints .in contrast to ref . , however , the moves are not stochastic , but rather deterministic , and constructed in such a way that they decrease the functional , and do this optimally .since the free energy landscape of this problem has a simple structure ( as discussed in sec . [ sec : variational_approach ] ) , the procedure relaxes the fields into the one and only minimum , which is the solution of the poisson boltzmann equation .the algorithm can be summarized as follows :  1 .distribute the fixed charges .2 .  classify the grid points .distribute the ionic species uniformly in the moveable nodes .4 .  initialize the electric field .perform the field moves for all the plaquettes ( smallest closed loops ) in the grid .perform the concentration moves for all pairs of adjacent moveable nodes .check if the changes in the functional caused by steps 5 and 6 are less than a given tolerance : if yes , then stop , otherwise , return to step 5 .  in the beginning ,fixed charges and ionic species must be distributed over the grid .fixed charges are usually associated with surfaces .therefore elements of surface charge density must be mapped onto elements of volume charge density , so that they can be associated with some of the nodes .these nodes are then marked as `` fixed '' and no particles can enter or leave them after initialization .further nodes may be marked as `` fixed '' if they are known to be empty ( for example , if they represent the interior of a particle ) .the nodes representing the volume where ions can move are marked as `` moveable '' .initially , each ionic species is uniformly distributed over the `` moveable '' nodes .choosing the correct amount of charges then automatically results in charge neutrality of the overall system .the next step consists in initializing the electric field so that it satisfies gauss law for the initial charge distribution .one possibility is to solve the poisson equation with some numerical method .this needs to be done only once , in the initialization .an alternative , based on charge neutrality , is to initialize each component by means of a recursion over the spatial dimensions , which is equivalent to applying gauss law to linear chains of nodes .first the lattice is decomposed into a set of planes perpendicular to the and it is required that takes the same identical value for all links with identical . then in one ( arbitrary ) plane of links we set .starting from there , we can then calculate step by step in the subsequent planes of links , where the change in is given by the plane averaged charge density between the links .assuming that the procedure is started at the charge plane , it reads where is the plane averaged charge density at .charge neutrality combined with the periodic boundary conditions ensures that this procedure will give consistent results after closing the one dimensional loop .then each plane is decomposed into a sequence of lines , perpendicular to the and the , and the analogous procedure is applied to obtain the field in .the charges which occur here are the line averages , where however the plane averages have been subtracted ( the latter have already been taken into account via ) .finally , the lines are decomposed into sites , and is determined from the remaining charges where both line and plane averages have been subtracted .  for field changes , elementary closed loops on the faces of the unit cells ( plaquettes )are considered . for the orthorombic lattice ,these are comprised of four nodes and respective links , such that each node is connected to two plaquette links .now , these four fields are modified in such a way that the flux on each link is changed by the same amount ( taking into account the orientation along the closed loop ) . therefore , gauss law will still be satisfied after that move , since at every node there will be some more flux entering but also the same amount of flux leaving .let us , for example , consider a plaquette perpendicular to the , with a sequence of fields , , , along the loop , where , are positive if the field points in positive ( and analogous for , ). then the field updates are given by or , in terms of fluxes , where the parameter can be chosen arbitrarily without violating gauss law .the associated change in the functional is given by which is minimized for this yields the optimal values for the field changes .axis , for various grid resolutions as indicated by the legends.,scaledwidth=47.0% ]  axis , for various grid resolutions as indicated by the legends.,scaledwidth=47.0% ]  for concentration moves between two adjacent nodes ( connected by a single link ) , gauss law is also conserved if the electric flux is updated accordingly .suppose that before the move the two adjacent nodes and have , respectively , concentrations and of some ionic species with valence .without loss of generality we may assume that node has a larger index value than node . the electric flux from node to node then given by , where is the length of the link , and the field on it .now , a certain ( positive or negative ) amount is moved from to , i. e. gauss law tells us that the flux should change by , and this is the case for the resulting change in the functional is given by in order to find the optimal value for , we minimize this expression .the result is a nonlinear equation , which must be solved numerically . by introducing the equationis transformed to which shows that it has exactly one solution ( the slope of the left hand side is always positive ) . since ,the solution will satisfy the condition which is equivalent to , , such that will be in the physically admissible range .finally , the shape of the left hand side guarantees that a newton iteration starting at will always converge ; hence , this rapid procedure was implemented .in this section , the feasibility of the present approach is demonstrated by two numerical examples .the choice of parameters is inspired by previous computer simulations on electrokinetics done in our group .we therefore quote them here in unscaled `` physical '' units , where denotes our elementary length scale ( the lennard jones diameter in our simulations ) .all calculations are done with a bjerrum length .the fixed charge distribution ( boundary condition ) consists of positive charges only , while there is only one ionic species , the monovalent counterions with valence .the resulting data are also given in `` physical '' units .consider that the fixed charges are distributed in two infinite parallel plates , perpendicular to the -axis , placed at and .the surface charge density in each plate is . furthermore , suppose that there are no salt ions and that the counterions ( of valence , concentration ) are distributed in the region between the planes .charge neutrality is given by   for this case , the problem is one dimensional and its analytical solution is well known .therefore , it is ideally suited to test the algorithm .in reduced units , the poisson boltzmann equation is which can be interpreted as newton s equation of motion of a particle with unit mass , whose coordinate is and where the time corresponds to .hence , this equation can be solved via standard methods of classical mechanics .the result is given by where the parameter is related to via .the surface charge density is given by therefore , can be obtained by solving a simple nonlinear equation numerically .  [ cols="^,^,^ " , ]   as one sees from tab .[ tab : comparison ] , the number of necessary iterations and the amount of cpu time are quite large .we have therefore looked for strategies to speed up the procedure without sacrificing the basic formulation that provides intrinsic stability .one possibility is to remove the rotational component of the electric field not by means of plaquette moves but rather by solving the poisson equation . within the chosen discretization scheme , the electrostatic potential needs to be an object associated with the sites , such that the electric field on a link is obtained by simply taking the potential difference between the adjacent sitesthis leads to the simplest possible finite difference scheme for the poisson equation , which can be solved efficiently and in a stable way by using a fast fourier transform ( fft ) and the appropriate lattice green s function .this has the advantage that one single lattice sweep not only reduces the rotational component ( as is the case for the plaquette moves ) but rather eliminates it completely .therefore the fft promises to increase the convergence speed .an easy implementation is possible using the well known and efficient fftw3 library routine .it should be noted that the link moves , which update the concentrations and the fields simultaneously , remain unchanged , such that the procedure still stays strictly on the constraint surface .  in a first implementation , we eliminated all plaquette moves and replaced them with fft sweeps done during initialization as well as subsequently after every 25th link sweep .as seen from tab .[ tab : speedup ] , this improves the efficiency roughly by a factor of .these results were obtained on the same computer as those of tab . [tab : comparison ] .furthermore , we can tackle the slowdown that comes from the fact that the ions have to be moved by site by site hops throughout the system ( `` hydrodynamic slowing down '' ) . to this end, we first run the calculation on a rather coarse grid ( in practice , we started with ) , such that most of the necessary `` mass transport '' is already done in that preliminary run .starting from there , we go to a finer grid ( in practice , we reduced the lattice spacing in all three directions by a factor of two ) and linearly interpolate the output of the previous run onto that grid . then the free energy is relaxed again ; the output of that run is interpolated onto a yet finer grid , and so on .obviously , this can be done rather easily by a straightforward recursion , until the desired grid resolution is reached .the runs before the finest resolution may then be viewed as a `` pre conditioner '' .this optimization yields another speedup by roughly 25 , as seen from tab .[ tab : speedup ] .these are obviously two rather simple optimizations , which do not interfere with the basic data structure of the simple cartesian grid .further optimizations , which are however much more complicated , are possible by ( i ) adaptive mesh refinement ( i. e. a fine resolution is only used in those regions where the fields vary strongly ) , and ( ii ) using finite element type unstructured grids. such more advanced approaches would be based upon constructing the dual ( voronoi ) lattice in order to define and calculate the fluxes .while this is expected to yield further substantial speedups , this was not attempted here , and is rather mentioned as a suggestion for the larger community .variational techniques were used to develop an algorithm for the poisson boltzmann equation .the required amount of memory scales linearly with the system size . in our first simple implementation ,all moves are local .this leads to fast memory access and fast calculations at each move .furthermore , the algorithm can be easily parallelized . on the other hand ,the charges need time to move between distant nodes .in fact , while the time spent on each move is essentially constant , the total number of sweeps required depends highly on how large a grid is swept .considerable speedups were possible by using ffts and a hierarchical pre conditioner , but the basic `` hydrodynamic slowing down '' remains still present .further speedups are likely to be possible by adaptive mesh refinement and by using unstructured grids .existing algorithms based on standard discretizations of the differential operators are probably significantly faster than even the fastest of our current implementations. nevertheless , the approach that we have outlined in the present paper has the inherent advantage that its mathematical formulation provides _ intrinsic _ stability .this is mainly due to the fact that the free energy functional is formulated in terms of the electric field instead of the electrostatic potential , which provides a search for a _ true minimum _ in function space , rather than for a saddle point . as a result, the algorithm is very robust , in the sense that it will _ always _ converge to the solution , and at every iteration it approaches the solution more closely .furthermore , the essential conservation laws that are at the heart of electrostatics conservation of mass and conservation of electric flux are built into the formulation with machine accuracy .we believe that these are fundamental advantages directly related to the underlying physics of the problem , and we consider them as much more important than implementation details .the maggs formulation provides a new way of thinking about electrostatics , and we hope that this , in combination with existing numerical `` technology '' , will in the future bring about very useful algorithms .

Abstract: a variational approach is used to develop a robust numerical procedure for solving the nonlinear poisson boltzmann equation . following maggs et al . , we construct an appropriate constrained free energy functional , such that its euler lagrange equations are equivalent to the poisson boltzmann equation . this is a formulation that searches for a true minimum in function space , in contrast to previous variational approaches that rather searched for a saddle point . we then develop , implement , and test an algorithm for its numerical minimization , which is quite simple and unconditionally stable . the analytic solution for planar geometry is used for validation . some results are presented for a charged colloidal sphere surrounded by counterions , and optimizations based upon fast fourier transforms and hierarchical pre conditioning are briefly discussed .


complex networks have emerged as an area of intense research activity in recent times .this is because they provide an effective framework for representing and modelling many types of phenomena occurring in complex systems .such complex systems arise in several diverse areas , ranging from neuronal systems to social sciences . in these cases ,the evolution of the network and hence the emergence of collective behaviour depend both on the nodal dynamics and the network structure or the topology of the connectivity among the nodes .earlier studies were mainly on static networks where the topology remains the same but dynamics at the nodes evolve under the topology giving rise to interesting dynamics like synchronization , clusters , amplitude death etc .recent reports indicate how rewiring mechanisms , where the topology is repeatedly changed , can enhance the richness of the dynamics .it is now well established that nodal dynamics and topology are equally important in real world situations and their interplay can lead to interesting results .a well defined network structure is often an important feature whose evolution can affect relevant biological processes . at the same time, the topology of connections obviously plays a role in information processing , spreading of diseases etc .however most of the network growth methods reported , do not rely on nodal dynamics , but depends on structural changes alone to produce desired results .several cases of synchrony and phase locking can be found in central nervous system and the resultant coherent activity plays an important role in behaviour and cognition . in this context ,phase locked cluster oscillations are observed in periodically forced neuronal networks and also in networks with coupling of inhibitory and excitatory nature .also it is reported how a fully synchronised single cluster breaks into two anti phase clusters due to phase resetting controls .co existent clustered states depending on initial conditions are shown to arise with inhibitory coupling .thus it is important to understand the mechanism responsible for the formation of phase locked clusters and other possible network structures and the role of nodal dynamics in stabilising them .  in this paper , we report how periodic attachment or removal of links based on a chosen strategy can establish coherent structures in a complex network .these emerge as the network evolves under both nodal dynamics and the proposed connectivity changes .we are motivated by the fact that real world networks often evolve with changes in topology in which links can appear or disappear as the whole system evolves .this is especially relevant in biological and social networks where contacts between systems are not constantly realised but occasionally activated . such strategic evolution of topology is shown to result in favourable outcomes in social networks .it is reported that activity dependent interactions in a network of phase oscillators can induce coherent state , two cluster state , and chaotic state .it has been shown that in the context of fitzhugh - nagumo ( fhn ) oscillators , rewiring the interaction pattern will induce collective firing such that lack of connectivity is compensated by dynamic rewiring .so also , adaptive coupling to enhance synchrony has been studied earlier .the interplay between dynamics and topology is highly relevant in deriving dynamical criticality and optimal properties . along these lines ,several evolutionary design algorithms have been introduced for coupled phase oscillators .it is also reported how functional sub networks can be evolved according to the network structure and dynamics .these studies indicate that rewiring or dynamic evolution enhances the scope and applicability of networks and result in interesting and richer dynamical states .  in our methodwe follow a strategic growth of the network where connectivity changes are done periodically using a condition that is based on relative performance as governed by the dynamics at the nodes .this method , while changing the total links at every re wiring , finally drives the topology to stable states and thus provides a very simple mechanism for growing networks and stabilising them to different topologies .we illustrate this in networks with fhn neuronal model as nodal dynamics . starting from a very sparse network ,we show how selective removal and adding of links based on this specific strategy can result in phase locked clusters in which nodes in each cluster are in almost synchrony , but are phase locked with nodes of the other cluster .it is interesting to see a stable bipartite structure emerging under such an evolution with two clusters anti phase to each other .moreover the specific strategy chosen here leads to a structure in which there are no links among synchronised nodes in the same cluster but all to all connections between the nodes in different clusters .we consider a large number of realisations with different starting initial conditions to establish that such an emergent topology is preferred by the system under the strategy used by us .we indicate how to establish the stability of the resultant topology , its dependence on the parameters in the design of the strategy and the role of nodal dynamics in this scheme .we consider the fitzhugh nagumo ( fhn ) neuron model as the nodal dynamics in our study .it is widely used to study the dynamics of neurons especially neuronal synchronization and desynchronization phenomena in the brain .the intrinsic dynamics of fhn systems is modelled by the following equation , here , represents the voltage of the neuron and represents the recovery variable .the variations in the voltage represent the spikes observed in neurons .this system also can represent the dynamics of any excitable system since it has two time scales , which can be controlled by choosing appropriate values for the parameter .  in the study reported in this paper, we consider a dynamic network of such fhn neurons coupled as the strength of diffusive coupling is given by .the network topology is decided by the elements the adjacency matrix . if , the neurons and share a link .we choose the matrix to be symmetric such that all couplings are bidirectional .further , self - couplings are avoided by choosing .we introduce a strategy for evolving the system in which in addition to the dynamical evolution of the nodes , a new network topology is selected after a rewiring time of .it is linked with nodal dynamics such that the strategy depends on the co - performance of nodes as indicated by the suitably defined distance between them in phase space .thus the rewiring is based on local information on nodes and not on global measures like order parameter or total number of links etc . to quantify this, we define the distance function between a pair of nodes , as the euclidean distance in phase space between them given by ( clearly ) .the steps involved in the strategic evolution of the network are as follows .  1 .start with a very sparse network of systems 2 .after a rewiring time of , check the value of at very pair of nodes .3 .  if , a chosen threshold , attach a link between and by setting .4 .  if , detach the link between and by setting . 5 .  repeat steps after very time interval .  thus in a waythe matrix itself is also co evolving with the nodal dynamics . here , the network is potentially global , which means that , any neuron can get connected to any other neuron , based only on the euclidean distance between them .the evolution is continued till the network topology stabilises . in the next section, we present the detailed analysis on the topological states of the network that result from the evolution scheme introduced .the analysis is done numerically by using the parameters , such that the intrinsic dynamics is periodic spiking . the typical reset time is chosen to be while the coupling strength is varied from to and the threshold is varied from to . for the numerical integration of eq .[ eq : fhn_net ] , we use adams - bashforth - moulton predictor - corrector method .we consider ten fhn neurons in random topology evolving under the strategy we have introduced . from numerical integration of eq .[ eq : fhn_net ] starting from random topology and random initial conditions , we find that , for certain initial conditions the systems stabilize to a -clustered state for suitable values of coupling strength .the time series from all the nodes for this state is shown in fig .[ fig : ts_10fhn_2clus ] .  .time series of the voltages of ten coupled fitzhugh - nagumo systems[ eq : fhn_net ] ) showing -cluster state for , and . in this case , both clusters have systems each . here ,the parameters of the individual systems are kept at and . ]it is clear that five of the nodes are in synchrony and are in anti phase with the remaining five nodes .thus this is a 2 cluster network with five nodes in each .as the network topology evolves with the system dynamics , it stabilizes to a bipartite structure with nodes in different clusters forming two groups . due to the specific strategy chosen, there is no connection among the nodes in synchrony within a cluster while all pairs of nodes from different clusters are connected .the resulting network structure is shown in fig .[ fig : bipart_example ] .   a representation of the stable bipartite network .blue and red nodes form two different groups ( clusters ) . here , both clusters have nodes each . note that there is no two blue nodes ( and no two red nodes ) share a link and that all pairs of blue and red nodes are connected . ]  the corresponding adjacency matrix is shown in fig .[ fig : bipart_example_adj ] . here , black squares represent ones and grey squares represent zeros .the adjacency matrix is symmetric since we consider all connections to be bidirectional .   adjacency matrix corresponding to the bipartite network shown in fig .[ fig : bipart_example ] . here , black squares represent ones and grey squares represent zeros .the adjacency matrix is symmetric since we consider all connections to be bidirectional .note that rows ,,, and are similar and that rows ,,, and are similar .this corresponds to the cluster structure seen in fig .[ fig : bipart_example ] . ]we proceed with the numerical study by choosing large number of different initial conditions and find that , different types of bipartite networks ( & , & , & , & and & ) are possible for different sets of initial conditions in phase space .we use the term -cluster state to refer to all these states collectively .there are also some initial conditions that evolve into a single cluster or a -cluster state , with all the systems in almost synchrony and the resulting topology is a set of disconnected nodes . by almost synchrony we mean ,as mentioned in the difference in the firing times of two systems is less than ten of their inter spike interval .these different topological states therefore multi stable states of the system .to identify all the possible resultant network topologies , we first find the number of clusters ( number of clusters for a bipartite network is , for a tri - partite network is etc . ) from the adjacency matrix as described below .then we find the number of systems in each cluster from the total links in the evolved topology .as mentioned above , the systems which are synchronized among each other are not connected and the corresponding elements in the adjacency matrix are zeroes ( ) .conversely , if then , systems at nodes and are not synchronized and thus , belong to different clusters .this means that , the number of clusters in each topology can be identified as the number of unique rows in the adjacency matrix . for example , in the adjacency matrix shown in fig .[ fig : bipart_example_adj ] , we find that there are only two unique rows ( or columns , since the matrix is symmetric ) namely , row- and row- .all other rows are replicas of these two unique rows .thus , we identify the network represented by the adjacency matrix shown in fig .[ fig : bipart_example_adj ] as bipartite .the -cluster can also be identified as the network with number of links , .as noted earlier , all pairs of nodes in different clusters are connected to each other .thus , the number of links in a -clustered state of ( & ) can be estimated to be , since the only node in the first cluster is connected to all nodes in the second cluster , and no two nodes in second cluster share a link .similarly , ( & ) clustered state has links , ( & ) clustered state has links , ( & ) clustered state has links , and ( & ) clustered state has links . in the case of -clustered states the number of unique rows in the adjacency matrix is found to be .the number of links in a -cluster state of ( , & ) can be estimated to be , since the only nodes in first two clusters make one link with each other and nodes each with the third cluster .similarly , ( , & ) clustered state has links , ( , & ) clustered state has links , ( , & ) clustered state has links , ( , & ) clustered state has links , ( , & ) clustered state has links , ( , & ) clustered state has links and ( , & ) clustered state has links .thus , each resultant topology can be identified from the number of unique rows in the adjacency matrix and the number of links .as we have noted earlier , we find our method results in multi stable topological states depending on their initial states .it is to be noted that by initial state we mean both the starting topology and the initial conditions in phase space for the nodes .the number of systems in each cluster or the variety of possible clusters also depends on initial starting states . to establish the multi stability and to get a comprehensive picture of stabilised topologies , we consider realisations of the nodal states and evolve them based on the method described in the previous section .for each stable -cluster and -cluster topology , we calculate the frequency of occurrence defined as , frequency = number of occurrences of a certain topology / total number of initial conditions .this can be understood as the probability of occurrence of each stable topology .this result is used to plot the histogram shown in fig .[ fig : hist_ep_2 ] .the parameters are chosen as , and . herethe red line at corresponds to -cluster state with links while the green lines correspond to different -cluster topologies .we find that the -cluster state has the maximum frequency of occurrence for this set of parameters , while the -cluster state does not stabilise .we identify the various cluster states from the number of links .thus green line at corresponds to bipartite cluster(, ) , that at corresponds to ( , ) , at to ( , ) , at to ( , ) and at to ( , ) cluster .  .frequency of occurrence of each network topology for a net work of ten fhn systems .the parameters are chosen to be , and .red line at corresponds to -cluster state with links .green lines correspond to different -cluster topologies . for details , see text . ]we repeat the analysis for a smaller value of the coupling strength and see that , then -cluster states are also present in addition to -cluster and -cluster states .the dynamics in these clusters is phase locked with each other .a histogram showing the frequency of occurrence of , and clustered state for is given in fig .[ fig : hist_ep_p2 ] . herewe choose red line at to represent -cluster state with links .the green lines in the figure correspond to different -cluster topologies and blue lines to various cluster states .we find that , the -cluster state with & nodes has the maximum frequency of occurrence for this set of parameters .green line at corresponds to ( & ) cluster .the various possible clusters within the cluster and cluster topologies can be identified in the figure as follows : green lines at corresponds to ( & ) cluster .green line at corresponds to ( & ) cluster .green line at corresponds to ( & ) cluster .green line at corresponds to ( & ) cluster .blue line at corresponds to ( , & ) cluster , at corresponds to ( , & ) cluster , at corresponds to ( , & ) cluster , at corresponds to ( , & ) cluster , corresponds to ( , & ) cluster , at corresponds to ( , & ) cluster , at corresponds to ( , & ) cluster and at corresponds to ( , & ) cluster .  .frequency of occurrence for each network topology for .the parameters are chosen to be , and .red line at corresponds to -cluster state with links .green lines correspond to different -cluster topologies and blue lines to various cluster states . for details of various possible sub types in and clusters , see text . ]thus we see that the network settles to different final topologies depending on parameter values , coupling strength and initial conditions .we refer to these as topological fixed points of the system since the network settles to a final topology which does not change further with time . to identify stable fixed - point topologies ,we evolve the coupled system from different initial conditions , for time steps .the data for the first time steps is neglected as transients .we take the time - average and the standard deviation of for each realization .if the standard deviation , , we consider the topology to be a topological fixed point .we mention that the fixed point nature is in the topology and not in the nodal dynamics since even in the final state , the nodes in the topological fixed point will have dynamics as given by eq .[ eq : fhn_net ] .the number of links in the different fixed point topologies remains constant in time .this is clear from fig .[ fig : ts_m ] , where we plot the number of links at different reset times .   a representative figure showing the topological fixed points .here the number of links is plotted against time for a network of ten fhn neurons at coupling strength and reset time .the value of is chosen to be .note that the time is shown in units of reset time , . ]however we find that while most of the initial conditions get stabilized to a -cluster or -cluster state , there are a few remaining initial conditions that do not stabilise to any fixed point topology . among them , especially for larger values of , we find an interesting behaviour of switching between topological states .this is illustrated in fig .[ fig : ts_m_switch ] by plotting the number of links as a time series against the rest times for .number of links against time for a network of ten fhn neurons for .the other parameters are kept fixed at the values given in fig .[ fig : ts_m ] . here also , the time is measured in units of reset time , .we see that the topology switches between a -cluster state of no connections ( ) and a bipartite state ( ) . ]  to further establish the stability of the observed topological fixed points , we start the numerical integration of eq .[ eq : fhn_net ] from a random network of ten fhn neurons for the set of parameter values , and .after time steps ( resets ) , we find that the system gets settled to a topological state , either a -cluster or -cluster state .to test the stability of this resultant topology , we disturb one link of the resultant topology ie .we choose a link at random and if the element corresponding to that link in the adjacency matrix is one , we change it to zero and vice - versa .then the system is evolved for some more time and we see that , the system goes back to its original topology in one or two reset times .we repeat the analysis for a different values of to confirm that the topology is stable against random perturbations .the results presented so far are done considering a random initial topology . to study whether the initial topology affects the results , we repeat the numerical analysis with different starting topologies . for example , we start from & bipartite cluster with and and get single cluster ( ) and various -clusters and -clusters as final topologies. this is repeated with all - to - all connected topology and -clustered ( ) topology as initial states , and still they end up in multi stable topological states .to check whether the results seen are the effects of the intrinsic dynamics of the systems or the strategy , we consider ten fitzhugh - nagumo systems in all - to - all coupled topology and let them evolve for time units with out applying the strategy discussed above .we find that , for sufficiently large coupling strengths ( ) , depending on initial conditions , synchronized states where all systems are in one cluster and clustered states where systems divide themselves into two clusters can happen .we find qualitatively similar results for the simulations starting with random network topologies .though the two clustered state coexisting with the fully synchronized state seems counter intuitive similar results has been reported in the case of two coupled lorenz systems .we next consider the limiting case of two systems . in this case also , depending on initial conditions , the two systems can evolve into in - phase or anti - phase states .we further study the evolution of these two systems in the presence of the strategy discussed above .the parameter plane showing regions of synchronization is shown in fig .[ fig : pp_random ] . in this case, we increase the value of from to in steps of and from to in steps of . for each pair and , we plot the euclidean distance between two neurons as colour coded plot .blue regions ( ) thus correspond to synchronized states .the coupling strength is kept fixed at .initial conditions are , , and .we have verified that , starting from this set of initial conditions , the systems do not get synchronized in the absence of strategic evolution .regions of synchronization in the parameter plane for two strategically coupled neurons . for each pair and , we plot the euclidean distance between two neurons as colour coded plot . ]this means that , by properly choosing the parameters and , the systems can be driven from one topology of anti phase state to the other of in phase state .hence , we infer that both the intrinsic dynamics and the strategic evolution have a role in stabilizing the topological states of the system .we investigate further the role of the most relevant parameters like , and in the outcome of stabilised topology . for thiswe do the full analysis for different values of and by keeping .we find that in general the relative frequency of occurrence for each topology varies for different values of parameters .the change in frequency of -clustered state for increasing is shown in fig .[ fig:1clus_freq_vs_k ] .we see that for all curves , the frequency of occurrence increases as is increased .  .frequency of occurrence of -cluster state for increasing values of coupling strength .different curves correspond to different values of . in all cases ,the frequency increases as is increased . ]  similarly , we illustrate this for the cluster state for the specific case of , cluster in fig . [ fig:2clus_5 ] .similar curves are obtained for other -clustered states .we infer that as increases , the frequency of occurrence of cluster state decreases .frequency of occurrence of , cluster for increasing .curves of different colours correspond to different values of .the frequency decreases for higher values of coupling strength . for ,this cluster occurs only for . ]we study the effect of varying on stabilisation of topologies . for thiswe compute the fraction of all the stable fixed point topologies for different values of . in fig .[ fig : stable_vs_thr ] , we plot this fraction for increasing values of .we see that , for very small value of ( ) , more than half of the initial conditions leads to unstable topologies . as is increased , the fraction of stable topologies is found to increase .we repeat the analysis for different values of coupling strength in the range and different values of reset times ( , and ) and the results are incorporated in fig .[ fig : stable_vs_thr ] .   fraction of stable fixed point topologies for increasing values of . here, we have considered neurons .red triangles represent , green circles represent and blue squares represent . for the case, the analysis is done only for .different points of the same colour represent different coupling strengths in the range increased in steps of for both and cases . for case ,the coupling strength is increased in steps of in the range . ]it is interesting to note that , there is an optimal range of , from to , where most of the initial conditions lead to topological fixed points as the frequency of stable topologies reaches almost .dynamic neuronal networks form an important paradigm of increasing importance in brain research , especially related to the functional analysis of biological neuronal networks . often in such contextsit is important to study the temporal dynamics in relation to spatial patterns . herewe present a very simple evolutionary strategy that can utilise the mutual effect of spatial and temporal factors to stabilise a particular topology in the network .the dynamic evolution studied is based on a rewiring scheme derived from local information on the co - performance of adjacent nodes .we find this facilitates the formation of bipartite structure with two phase locked clusters in addition to a prevalent single cluster .we analyse the probability of occurrence of different possible bipartite topologies in a given network for specific chosen values of the parameters . considering the relevant parameters involved in the process as , and , we study in detail the effect of varying them on the final stabilised topology .some of the important results and conclusions are given below .for large value of , the single cluster is mostly preferred , while smaller values of lead to bipartite and even tripartite topologies . for large value of the threshold ,switching between topologies occur .there is an optimal range of in which almost all initial conditions lead to topological fixed points .the dynamics at the nodes has a role to play in the evolution of fixed topology . in a network of same systems without any strategyalso phase locked clusters are possible .what the strategy does is to drive the network to a stable topology and keep it there without any links within a cluster . in the limiting case of two systems , we find the strategy can change anti- phase to in - phase and vice versa .this means strategy can induce a tendency to settle to definite topologies .conceptually , we feel the topology to which the network stabilises is similar to stable fixed point in dynamical systems . accordingly, we establish the stability by considering small perturbations from the stable or fixed point topology .even though multi stability for dynamical states have been reported for networks and other coupled systems , to the best of our knowledge , this is the first time multi stability in topology is studied in detail .the homogeneous model of network studied to establish the growth of phase locked clusters may not capture all the complexities of neuronal dynamics or real world networks .however , it opens up the possibility of keeping the emergent state of a network in synchronization without any instantaneous links between their nodes but by keeping a strategy alive .also the state is really robust since the moment small perturbations or noise arise due to external effects etc .the strategy will become active and change topology accordingly to retain the synchronization .this need based re wiring can be optimised for minimum energy expenditure in engineering systems or real world networks with possible biological justifications .emergence of bipartite structure is interesting and is concurrent with the presence of motifs with fan like structure in the network .usually bi partite networks are considered in relation to two different types of nodes .here we show how bipartite nature emerges by dynamic evolution from a single type of nodes .this can be considered as an extreme case of the conventional bi partites .it is well established that self organised structures can emerge from a collection of interacting elements .the study reported here confirms that a strategy based dynamic evolution can drive the system to definite structures and stabilised topologies .thus this provides one more instance that the co - 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Abstract: we report how strategic evolution can stabilize topological states in a network of fitzhugh - nagumo systems . the evolution follows a repeated process of adding or deleting of links between two nodes that is decided based on a threshold set for the closeness between them . this results in a need based rewiring that keeps the systems in synchrony in a single cluster or in phase locked states in different clusters . we analyse in detail the occurrence of such multi stable topological fixed points with corresponding emergent dynamical states using the frequency of occurrence of each state when evolved from a large number of initial states . we find that there is an optimum range for the threshold used in the strategy that results in maximum frequency for the stable topological states .