The dendritic tree contributes significantly to the elementary computations a neuron performs while converting its synaptic inputs into action potential output. Traditionally, these computations have been characterized as both temporally and spatially localized. Under this localist account, neurons compute near-instantaneous mappings from their current input to their current output, brought about by somatic summation of dendritic contributions that are generated in functionally segregated compartments. However, recent evidence about the presence of oscillations in dendrites suggests a qualitatively different mode of operation: the instantaneous phase of such oscillations can depend on a long history of inputs, and under appropriate conditions, even dendritic oscillators that are remote may interact through synchronization. Here, we develop a mathematical framework to analyze the interactions of local dendritic oscillations and the way these interactions influence single cell computations. Combining weakly coupled oscillator methods with cable theoretic arguments, we derive phase-locking states for multiple oscillating dendritic compartments. We characterize how the phase-locking properties depend on key parameters of the oscillating dendrite: the electrotonic properties of the (active) dendritic segment, and the intrinsic properties of the dendritic oscillators. As a direct consequence, we show how input to the dendrites can modulate phase-locking behavior and hence global dendritic coherence. In turn, dendritic coherence is able to gate the integration and propagation of synaptic signals to the soma, ultimately leading to an effective control of somatic spike generation. Our results suggest that dendritic oscillations enable the dendritic tree to operate on more global temporal and spatial scales than previously thought; notably that local dendritic activity may be a mechanism for generating on-going whole-cell voltage oscillations.
The dendritic tree contributes significantly to the elementary computations a neuron can perform, both by its intricate morphology and its composition of voltage-gated ionic conductances [1]. Such active conductances can underlie a wide variety of dynamical behaviors, amongst others dendritic spikes and ongoing oscillations of the dendritic membrane potential [2]. Such active dendritic phenomena have been suggested as mechanisms endowing single neurons with significant computational power [3] and flexibility in the way the dendritic tree processes its inputs: whether as a global element, effectively collapsing the tree into a single functional compartment or with various parts of the tree acting as independent processing elements [4], [5]. While the possibility of powerful and flexible dendritic processing has been of great interest, the precise conditions under which dendrites can act independently or globally remain largely to be determined. In this report we address this key question, focusing specifically on the case where active properties lead to sustained intrinsic membrane potential oscillations in the dendrites. We develop a theoretical formalism, allowing for a succinct yet powerful description of the dendritic tree dynamics and yielding conditions under which the tree acts as a global oscillatory unit and how such action in turn controls spiking responses of the neuron. Membrane potential oscillations have been demonstrated in various types of neurons. Prominent intrinsic subthreshold oscillations have been found in stellate cells from entorhinal cortex layer 2 [6], [7], neurons from the frontal cortex [8], neurons from the amygdala complex [9], [10], and pyramidal cells and interneurons from the hippocampal CA1 area [11], [12]. Although these membrane potential oscillations are normally recorded at the soma and thus are considered to be of somatic origin, several lines of evidence suggest dendritic loci of generation. First, many of the conductances thought to underlie the generation of such oscillations reside predominantly in the dendrites, sometimes specifically in the distal parts of the dendritic tree. For example, in the apical dendrites of hippocampal CA1 pyramidal neurons, the density of increases strongly with distance from the soma [13], and reaches very high values in the thin distal branches [14]. Second, several studies have suggested the existence of clusters of ionic conductances that are responsible for the generation of dendritic spikes [15]. While most of the direct electrophysiological evidence regards excitable behavior, demonstrating the generation of dendritic spikes in response to sufficient levels of depolarization, mathematical analysis has shown that neural membranes exhibiting excitability can readily pass to oscillatory regimes in an input-dependent manner (e. g. see [16]). Third, in several cases, oscillations have been directly recorded in dendrites. For example, recordings from hippocampal CA1 pyramidal neurons have demonstrated ongoing oscillations in the dendrites that include repetitive dendritic spikes, presumably involving Ca currents [17]. Furthermore, significant intrinsic dendritic oscillations have been observed in several neuronal preparations that depended on the interplay between the non-linear properties of NMDA synaptic receptors and intrinsic voltage-dependent currents [18], [19]. Crucially, while the onset of these oscillations was conditional on the activation of the NMDA synapses, the oscillations themselves were produced by mechanisms that were intrinsic to the postsynaptic cell and not by periodically structured synaptic inputs. Since NMDA receptors are largely localized on dendritic spines, and are hence electrotonically removed from the soma, these data may also argue for a non-uniform and local dendritic generation of membrane potential oscillations. Taken together, these experimental results suggest that dendritic trees can function as oscillators, perhaps conditional on the level of background depolarization or the presence of neuromodulators [20], while leaving open the question whether global cell-wide voltage oscillations could result from local dendritic mechanisms that are intrinsic even to distal dendrites and hence only weakly coupled to the soma electrotonically. Indeed, multiple intrinsic dendritic oscillators have been proposed to underlie the recently discovered intricate firing pattern of entorhinal grid cells [21]–[23]. This influential model suggests that the functional responses of entorhinal neurons recorded in behaving animals are a direct consequence of the generation of independent oscillations that are intrinsic to individual dendrites. Hence, this model presupposes the existence of multiple oscillators that are integrated at the soma, leading to the questions of how such dendritic oscillators may interact with the soma and with each other, and what sorts of collective behaviors the electrotonic structure of the dendritic tree might impose on the oscillations. In this paper, we study the dynamics of such interacting oscillators and their impact on signal propagation in single neurons, using mathematical analysis corroborated by numerical simulations of biophysical models. We treat the dendritic tree of a neuron as a network of oscillators coupled by stretches of relatively less active cable. This prompts us to combine two analytical methods: weakly coupled oscillator theory and cable theory. The theory of weakly coupled oscillators has been extensively used previously to study synchronization of multiple oscillators residing in separate cells interacting through synapses or gap junctions [24]. Since we focus on intradendritic oscillators which are continuously coupled via the membrane voltage, we use cable theory [25] to compute their interactions. We find that intradendritic oscillations can exhibit complex patterns of phase-locking. We characterize how this phase-locking depends on the intrinsic properties of the oscillators and on the membrane properties of the segment connecting them. Finally, we demonstrate how input to the dendritic oscillators can control the phase-locking and how in turn the phase-locked configuration can control somatic spike generation. These results provide a rigorous mathematical framework for the study of interacting dendritic oscillations that can be applied in the future to specific systems of interest, and also point to ways in which such oscillations can be utilized for non-trivial single cell computations.
The basic behavior of the system can be most easily understood by examining a simplified situation where the oscillators have a phase response curve that is approximately sinusoid and the perturbations from the cable are also nearly sinusoidal (e. g. when the oscillators are subthreshold with simple sinusoidal voltage traces). Hence the first Fourier component dominates in both and. The interaction function is then (6) where is a positive coefficient characterizing the strength of the coupling (see Equation 22 in Methods). The term gives the effective phase delay in the interaction between the two oscillators (figure 1Aii). In this term depends on the properties of the oscillators and summarizes the effect of cable filtering. It depends on the properties of the dendritic cable: , , and (see Methods). Using Equation 5 it is easy to show that the evolution of the phase difference between two identical oscillators is given by (7) The fixed points are the in-phase solution and the anti-phase solution (figure 1Aiii). The stable phase-locked solutions are those fixed points where the derivative of Equation 7 with respect to is negative: (8) The synchronous solution is thus stable when. When this solution is stable the anti-phase solution is unstable and vice versa. Notice that if we fix the properties of the oscillators, the constant is fixed. Then the value of uniquely determines which is the stable state (figure 1Aiv). Hence, to understand how the dendrite behaves as a function of the key properties of the cable we need only to look at how these affect. In the next sections we describe the behavior of with the consequent effect on phase-locking. The explicit expressions for the scaling of with the various parameters considered below are given in the Methods. In the previous section we limited our description and analysis to oscillators with a nearly sinusoidal phase response curve that receive perturbations which are also sinusoidal. In this way we could demonstrate how the parameters that define the oscillator and cable properties affect the phase-locking behavior of the system. However, as consequence, we only obtained and analyzed symmetric interaction functions. For such coupling functions, only the in-phase and anti-phase solutions are possible of which one is stable and one unstable. When and cannot be well approximated by a single Fourier component we need to take into account higher order terms. Including more Fourier components is likely to lead to asymmetry or skew of and, as we will show next, this affects the possible phase-locking behaviors. Above we developed a framework for analyzing the behavior of local oscillators embedded in the dendritic tree. Now we turn to the question of how such oscillating dendrites respond to inputs and impact the output of the neuron. We will show that the external synaptic input can control the phase-locked configuration of the dendritic oscillators and that this phase-locked configuration can then be transmitted through patterning of the cell' s action potentials. While a thorough analysis is beyond the scope of the present study, we give several salient illustrative examples using a model with a branched oscillating dendritic tree and a spike-generating soma. More specifically the model consists of a passive branching dendritic compartment with two Morris-Lecar type II oscillators at its two distal ends and an excitable soma that, for simplicity, we describe with an integrate and fire mechanism (figure 7A). Above we showed that under certain conditions, depending on the skew of the interaction function, the dendritic tree can be in a phase-locking regime where two stable phase-locked states co-exist (see figures 3C and 4C). In such a bistable regime, well-timed inputs to one or more dendritic oscillators can switch the locking between in-phase and anti-phase. Clearly, the membrane potential fluctuations at the soma depend on whether the dendritic oscillators are synchronized or not. In our model, they are largest in amplitude when the dendritic oscillators are in-phase. The soma can show this difference with its spiking pattern when such large amplitude fluctuations are supra-threshold, while smaller fluctuations (e. g. with asynchronous oscillators) are not. In figure 7 we illustrate the above mechanism. The initial parameters are such that both the in-phase and anti-phase state of the dendritic oscillators are stable (black dotted line in figure 7C). Oscillators starting from an initial phase difference move into the synchronous phase-locked state (red curve in figure 7B). This consequently leads to repetitive somatic spiking (blue traces in middle and bottom panel). A brief depolarizing current pulse to one of the oscillators (see black trace in top panel of figure 7B) moves them into the anti-synchronous state and the somatic spiking ceases. A subsequent synchronous current pulse to both dendritic oscillators can switch them back into the synchronous state and hence restart the spiking. Note that all the stimuli here are excitatory, yet depending on their timing, they can have a net excitatory or inhibitory effects on the cell' s spiking. We have also hinted, in a previous section, at another mechanism by which inputs to the dendrites can affect the phase-locked state. The input amplitude can change the oscillator frequency which in turn has an effect on the stability of the phase-locked state (see figure 1D). In figure 7B at time sec we increase the amplitude of the current input impinging on the oscillators which causes the system to move out of the bistable regime. The synchronized state loses stability and the oscillators gradually move into anti-phase locking. As a result, the soma stops spiking (at time sec). Note that the electrotonic separation between the oscillators remains constant (black dotted line in figure 7D) but that the bifurcation diagram itself changes. In turn, a decrease in the excitatory input would reinstate spiking. Hence, this mechanism allows the cell to encode an inverse of the input amplitude, or the inverse of the excitatory input rate.
The question of how local cellular processes may lead to global behavior has been of great interest for some time, in particular with respect to the signal propagation in extended structures such as the dendritic trees of cortical neurons. One of the aspects that remains a subject of active debate, is the dendritic mechanisms that ensure that local inputs on the dendrites – and in particular on the distal dendrites – have an impact on the global signal processing in the cell and ultimately on spike generation. We addressed this key question focusing specifically on the case of oscillatory dendrites. Thus, we studied the dynamics of dendrites that show intrinsic oscillations due to active voltage-dependent currents that present strong spatial inhomogeneities, hence leading to discrete oscillatory segments. Our prime question was to understand how global dendritic behavior, in this case the phase-locked oscillations, can arise from interactions between such local oscillators. To do so we developed an analytical framework to describe and understand the behavior of interacting dendritic oscillators and their impact on signal propagation within a neuron. Our goal was to understand when the oscillators within the dendrite would lock and hence the whole dendritic tree would act as a single oscillatory unit. Using the weakly coupled oscillator framework we have identified the requirements for the various phase-locking regimes of the dendritic oscillators. We characterized how the type of phase-locking depends on the intrinsic properties of the oscillators as well as on the membrane properties of the dendrite segment connecting them. We find that a central parameter in determining the phase-locked solutions is the electrotonic distance between the oscillators. This distance determines how strongly the dendritic cable filters the interactions between the oscillators, thereby determining the delay between the interactions. As a function of the electrotonic distance the phase-locking of identical oscillators alternates between in-phase or synchronized solutions and anti-phase solutions. We also showed how the phase-locking is affected by the presence of voltage-dependent conductances in the cable that connects the oscillators. Using the quasi-active approximation of the cable [29], [30] we found that the dependence of the stable phase-locked solution on the electrotonic distance is typically amplified by regenerative conductances (i. e. ionic conductances that amplify a voltage perturbation), whereas it is counteracted by restorative conductances (i. e. ionic conductances that counteract voltage perturbations) (see also [28]). It should be noted that the linearization of the active conductances in the dendrites is appropriate for small amplitude oscillations in the dendrite and is therefore in general a better approximation for subthreshold oscillations than for spiking oscillators. The mathematical approach that we used, builds on several studies which focused on the interaction between two neurons with repetitively spiking somata that interact via inputs at the dendrites [26]–[28]. A crucial difference with these studies is that rather than coupling via discrete synaptic events, we treat continuous coupling between the oscillators via the current-conducting cables. One consequence of the continuous coupling is that one needs both the phase response function and the voltage trajectory of the oscillators in order to compute the interaction functions and ultimately the phase-locked solutions. By computing the convolution of the voltage trajectory and the phase response function, which yields the interaction function for directly coupled oscillators, it is possible to get some insight into the types of phase-locked solutions that can be expected. The skew of the interaction function can show whether regimes can be expected in which both in-phase and anti-phase solutions are stable. Both the voltage trajectory of an oscillator and its phase response function can be determined numerically from a model of an oscillator and, at least in principle, also experimentally (see, for example, [34]). In the final section of our study we demonstrated how inputs to the dendritic tree can set the phase-locked state and how in turn the phase-locked configuration can control somatic spike generation. The first can for instance be accomplished by changing the frequency of the oscillators with the external input. The soma can subsequently detect the amplitude of the membrane potential fluctuations since this is affected by the phase-locked configuration. The time scale at which the dendritic oscillators move from one solution to another is set by the strength of the interactions between the oscillators. This time scale can be much longer than that of the different components of the system, e. g. the membrane time constant or the period of the oscillators. In this way, the phase difference between the oscillators can function as a memory. Related ideas have been previously discussed by Huhn et al [35]. We also showed that in the bistable phase-locked regime the state of the dendrites is easily set by transient inputs and “read-out” by the soma. This also can endow the neuron with a memory since brief external inputs can switch the neuron from a spiking to a quiescent mode and vice versa. Interestingly we showed that both the turn-on and turn-off signals (inputs) can be excitatory, their final effects defined by their timing. The focus of our report is complementary to that of a recent theoretical study of the subthreshold oscillations in the dendrites of mesencephalic dopaminergic neurons [36]. As these cells do not show any indication of distinct dendritic oscillators, the whole cell was modeled as one continuous oscillator with gradients in oscillator properties along the dendrites. Moreover, since there were no distinct oscillators, in their analysis Medvedev and colleagues assumed strong voltage coupling between neighboring compartments, enforcing synchronized oscillations throughout the cell. In contrast, our approach assumed weak coupling between the dendritic oscillators. This would not be appropriate for a spatially continuous oscillator. However, it is not possible to state in general at what precise electrotonic distance between two oscillators the weak coupling assumption becomes valid, since it depends on the strength of the interaction currents with respect to the intrinsic currents of the oscillators. However, our numerical simulations for a dendritic cable without the assumption of weak coupling, show that the phase-locking behavior of Morris-Lecar oscillators is consistent with weak coupling. One of the aims of the present paper was to set up an analytical framework for studying interacting dendritic oscillators. This opens up a wide range of questions that were outside the scope of the present study. For example, we focused our analysis on identical oscillators, while it is likely that dendritic oscillators will vary in their properties throughout the dendritic tree. For example, the diameter of the dendrites, which typically becomes smaller with increasing distance from the soma, can affect the intrinsic frequency of the oscillators. A gradient in the frequency of distinct oscillators is likely to lead to more complex phenomena such as traveling waves (see, for example, [37]). In fact the major focus of our study is to explore how local dendritic mechanisms may lead to oscillations expressed globally in the cell and hence visible at the soma, for example in somatic intracellular recordings. Our analysis showed that even electrotonically far removed dendritic oscillators can lead to voltage oscillations that significantly affect the soma voltage and hence spike generation. This suggests several experimentally testable predictions. In one possible experiment one can take advantage of imperfect space clamp in a electrotonically extended neuron. As a proof of principle, in a neuron where the oscillations are generated distally in the dendritic tree, voltage clamping the soma would not block such oscillations, and these should be seen in the current necessary to hold the somatic potential. In fact, results from [18] point in this direction, where in chick spinal cord neuron NMDA-dependent intrinsic oscillations were not blocked by somatic voltage clamp. A further prediction stems from the weak coupling between active dendrites. If active oscillations, such as periodically generated dendritic spikes, are generated in different segments of the dendritic tree, our analysis predicts that such spikes should interact and should exist in a stable phase-locked configuration, e. g. synchrony. Hence, should one of the dendritic segments be phase-shifted, such perturbation should 1. propagate to the other segment (the other segment should be phase reset) 2. the dendritic spikes should return to the phase-locked configuration 3. the time scale of this return should be relatively long and determined by the electrotonic distance between the active segments. While difficult such experiments are possible using the multiple dendritic recording techniques, such as those developed by Davie et al [38] in Purkinje cells. A recent model for the grid field properties of the entorhinal cortex layer II stellate cells [21], [22], [39] relies precisely on the ingredients considered in the present study. The model assumes that different dendritic branches emanating from the soma of these cells function as distinct oscillators. The oscillations are modulated by external inputs and the interference of the oscillators eventually determines the somatic spiking. Crucially, the model assumes that the dendritic oscillators operate independently. At a first glance, our results appear to argue against this: the various oscillators should phase-lock (hence lose their independence) even when the mutual coupling is weak. However, in principle, the locking may be slower than the behavioral time scale, allowing the oscillators to act quasi-independently on the behavioral time scale. Our analysis provides the appropriate framework to examine these issues: the scaling of locking in time and the biophysical implementation of grid-field formation via dendritic oscillators. Above we studied relatively simple cell geometries, however these form basic building blocks for more complex dendritic trees. Thus our framework should be valid for understanding global voltage oscillations in more realistic models of spatially extended cells. We would like to emphasize at this point that our general framework should also hold when – in addition to the distinct oscillators distributed throughout the dendritic tree – also the soma is regarded as an oscillator. These and other issues will be addressed in future publications. The framework we have developed, builds on the extensive mathematical theory of coupled oscillators and nestles nicely below the complexity of full compartmental models of neuronal dendritic trees. Yet our framework is sufficiently powerful and clear to both take into account certain key aspects of the dendritic tree structure and to be amenable to theoretical analysis of the dynamics of active dendrites and the computational function of such dendritic structures. These remain an active focus for further investigations.
We analyze the behavior of a system of two oscillators that are coupled via a cable. For this we need to compute the interaction between the two oscillators. Our approach is as follows. The oscillators provide the periodically forced end conditions for the cable equation. Assuming weak coupling the phase difference between the oscillators does not change significantly within one period of the oscillation. Thus we can solve the cable equation with such boundary conditions and leave the phase difference as a free parameter. In turn, the solution of the cable equation yields the currents flowing into and thereby perturbing the two oscillators at its ends. We let denote the membrane potential (in millivolts) along the cable at position (in centimeters) and at time (in milliseconds). The passive properties of the cable are determined by a membrane time constant (in milliseconds) and a length constant (in centimeters). The cable also expresses a voltage-dependent conductance with a gating variable with activation function and time constant (in milliseconds). The equations governing the membrane potential and the gating variable along the cable (excluding the oscillators) are (9) where is the leak reversal potential, is the reversal potential of the active current, and is the ratio of the maximal conductance of the active current to the leak conductance. The two oscillators form the periodically forced end conditions of the cable: (10) with and being the voltage traces of the two oscillators A and B that evolve according to (11) where is the membrane capacitance (in F/cm), is the leak conductance (in mS/cm), summarizes the voltage-dependent membrane currents generating the oscillations with the vector of gating variables given by standard kinetic equations (e. g. see Equations 28 and 29). The terms describe the perturbing currents from the cable to each oscillator with the small parameter denoting the coupling. For a cable with diameter (in cm) and oscillators with membrane surface area (in cm), , where is the intracellular resistivity of the dendritic cable (in kcm). The functions are given by (12) Consider identical oscillators when both and are dominated by the first Fourier component. One can show that the interaction function is given by (21) where is a positive coefficient, is a constant resulting from the cable filtering, is a constant that results from the specific properties of the oscillators and is a constant (see figure 1A). The expressions for the parameters are (22) (23) (24) (25) where and are, respectively, the absolute value and the angle of the complex number. The shape of the interaction function is determined by Equations 17,18 and 20. When the electrotonic separation between the two oscillators goes to zero, we have a system of directly coupled oscillators and the interaction function reduces to (27) where the constant. Introducing an electrotonic separation between the oscillators changes the shape of as a result of the cable filtering. When substituting Equation 18 into Equation 20 one sees that the symmetry of can only be affected by the -dependent term involving the voltage trace of oscillator B. As increases, the increasing cable filtering – determined by the absolute value of the term in Equation 18 – leads to dominance of a single Fourier component. Note that it is not necessarily the first Fourier component that will dominate. When a higher order Fourier component can be the dominant one. The equations for the Morris-Lecar type II oscillator [32] with parameters as in [40] read (28) with F/cm, mS/cm, mS/cm, mS/cm, mV, mV, mV, , A/cm, and where, , and. The equations describing the subthreshold oscillator are of the same form as those used by Morris and Lecar [32]. The oscillatory dynamics emerge from the interaction between the persistent sodium current and the hyperpolarization activated inward current. The current descriptions are based on the data from [33], [41]. The dynamics of are described by a single gating variable with activation function and time constant (in milliseconds). The voltage-dependent activation of is described by and is instantaneous. The equations read (29) with F/cm, mS/cm, mS/cm, mS/cm, mV, mV, mV, , A/cm, and where, , and. The numerical simulations for figure 4,6 and 7 used Morris-Lecar type II oscillators and simulations for figure 5 used the subthreshold oscillator model described above. The cable was discretized into isopotential compartments with electrotonic length. The perturbing currents from the cable to, for example, oscillator A are of the form with and denoting the membrane potential of the first two compartments. The parameter determines the coupling between the cable and the oscillators and is specified in the different figure captions. Simulations for figure 7 include a soma with an integrate and fire mechanism with a fixed threshold at mV. When the threshold is reached a spike is generated with a 1 ms peak at 30 mV after which the somatic is reset to mV for 4 ms. The phase response curves were calculated by determining the system' s adjoint [24].
A central issue in biology is how local processes yield global consequences. This is especially relevant for neurons since these spatially extended cells process local synaptic inputs to generate global action potential output. The dendritic tree of a neuron, which receives most of the inputs, expresses ion channels that can generate nonlinear dynamics. A prominent phenomenon resulting from such ion channels are voltage oscillations. The distribution of the active membrane channels throughout the cell is often highly non-uniform. This can turn the dendritic tree into a network of sparsely spaced local oscillators. Here we analyze whether local dendritic oscillators can produce cell-wide voltage oscillations. Our mathematical theory shows that indeed even when the dendritic oscillators are weakly coupled, they lock their phases and give global oscillations. We show how the biophysical properties of the dendrites determine the global locking and how it can be controlled by synaptic inputs. As a consequence of global locking, even individual synaptic inputs can affect the timing of action potentials. In fact, dendrites locking in synchrony can lead to sustained firing of the cell. We show that dendritic trees can be bistable, with dendrites locking in either synchrony or asynchrony, which may provide a novel mechanism for single cell-based memory.
neuroscience/neuronal signaling mechanisms
neuroscience/neuronal and glial cell biology
neuroscience/theoretical neuroscience
The Role of Ongoing Dendritic Oscillations in Single-Neuron Dynamics