Coarse-Grained Clustering Dynamics of Heterogeneously Coupled Neurons
© S.J. Moon et al.; licensee Springer 2015
Received: 24 June 2014
Accepted: 21 November 2014
Published: 12 January 2015
The formation of oscillating phase clusters in a network of identical Hodgkin–Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the synaptic coupling characteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscillatory single-cluster state can transition—through (possibly perturbed) period-doubling and subsequent bifurcations—to a variety of multiple-cluster states. The clustering dynamic behavior is computationally studied both at the detailed and the coarse-grained levels, and a numerical approach that can enable studying the coarse-grained dynamics in a network of arbitrarily large size is suggested. Among a number of cluster states formed, double clusters, composed of nearly equal sub-network sizes are seen to be stable; interestingly, the heterogeneity parameter in each of the double-cluster components tends to be consistent with the random variable over the entire network: Given a double-cluster state, permuting the dynamical variables of the neurons can lead to a combinatorially large number of different, yet similar “fine” states that appear practically identical at the coarse-grained level. For weak heterogeneity we find that correlations rapidly develop, within each cluster, between the neuron’s “identity” (its own value of the heterogeneity parameter) and its dynamical state. For single- and double-cluster states we demonstrate an effective coarse-graining approach that uses the Polynomial Chaos expansion to succinctly describe the dynamics by these quickly established “identity-state” correlations. This coarse-graining approach is utilized, within the equation-free framework, to perform efficient computations of the neuron ensemble dynamics.
A network of oscillators can form sets of sub-networks oscillating with phase lags among them ; these are often referred to as phase clusters. The dynamic behavior of such states has been investigated experimentally in globally coupled photochemical oscillators [2–4]. Certain features of the cluster dynamics have been studied numerically and/or theoretically in a variety of contexts: in arrays of Josephson junctions , in networks of inhibitory reticular thalamic nucleus (RTN) neurons , in phase oscillators with nearest neighbor coupling [7, 8], in models for synthetic genetic networks , and for identical Hodgkin–Huxley (HH) neurons with homogeneous, weak coupling , to name a few.
In this paper we study a specific type of clustering dynamics observed in synaptically all-to-all coupled networks of identical HH neurons, but for which certain synaptic coupling parameters slightly vary across the population, thus making the whole network heterogeneous. The main feature underpinning the clustering dynamics is (approximate) symmetry. Aronson et al., taking a group-theoretic approach, study the bifurcation features in oscillator networks with permutation symmetry, for an array of globally coupled homogeneous Josephson junctions . Dynamical systems with such permutation symmetries are known to give rise to a large number of coexisting states—symmetrically related to one another—which is referred to as “attractor crowding” [11, 12]. Hansel et al. show that multiple Fourier mode interaction terms are necessary in a phase-reduced model of homogeneously all-to-all coupled identical HH neurons in order to account for multiple-cluster formation . A generalization of the analytical framework for the Kuramoto-like coupled phase oscillators, including higher Fourier modes, has been attempted . The dynamical nature of the transitions between different cluster states in phase-reduced oscillator models and slow switching along the heteroclinic orbits involved have been discussed [3, 14–19].
In these studies of cluster dynamics, it is often assumed that the constituting entities are identical and/or the coupling strength is weak, allowing dimensional reduction via complete synchronization within each cluster and/or through the phase reduction procedure. An actual population of neurons (or more generally, oscillators), however, would hardly be expected to satisfy this homogeneity assumption. In practice, heterogeneity often exists inevitably, and it can have consequences for the collective dynamics [20–23], which are not easily deduced from the dynamics in the homogeneous limit. In the presence of weak heterogeneity, it is natural to expect that similar oscillators (characterized by neighboring values of the heterogeneity parameter) may trace similar dynamical trajectories and tend to belong to the same sub-network, when the network splits into coherent sub-networks. However, the HH neuron networks we study here do not follow this intuitive expectation. The heterogeneity parameter value does not determine which sub-network the corresponding neuron would belong to; the heterogeneous parameter (which is drawn from an i.i.d. random variable) within each sub-network is statistically consistent with that of the full ensemble heterogeneity distribution.
We demonstrate a coarse-graining approach enabling the analysis of the low-dimensional dynamics of single- and double-cluster states, which provides an efficient way of studying the coarse-grained clustering dynamics of an arbitrarily large network. This work extends the approach introduced to study coarse-grained single cluster dynamics of networks of heterogeneous Kuramoto oscillators . The approach is based on the Polynomial Chaos (PC), also known as Wiener’s chaos expansion , originally introduced to model stochastic processes with Gaussian random variables using Hermite polynomials; it has been further developed and widely used for uncertainty quantification . The PC-based approach utilizes the correlations that rapidly develop between the heterogeneity parameter values and the oscillator state variables. The same types of “identity-state” correlations are commonly observed to develop in a range of coupled oscillator models, including yeast glycolytic oscillators , van der Pol oscillators , and simplified neuron models .
The paper is organized as follows: The model and the parameter values used in it are described in Sect. 2, and some observations on the clustering dynamics in networks of heterogeneously coupled neurons are presented in Sect. 3. As a basis for understanding the dynamics of larger networks, the individual-level dynamics of a small number of neurons are analyzed in some detail (Sect. 4). A short survey of the PC expansion is provided and the dynamic behavior of large networks of neurons is studied in Sect. 5, while the derivation and the exploitation of our coarse-grained description of the clustering dynamics, utilizing the PC expansion, is presented in Sect. 6. The paper concludes with a brief summary and discussion.
We use almost the same parameter values as in , which correspond to a squid axon’s typical values at 6.3 °C: ; ; ; ; ; ; . The units for the parameters remain the same throughout the paper, and they are omitted hereafter, unless ambiguous.
The sigmoid is chosen to be ; its exact functional form does, to some extent, affect the overall dynamics. The network of neurons we consider is heterogeneous in the following sense: each neuron has a different synaptic time constant in Eq. (3). So even though the neurons are identical, they are coupled in a heterogeneous fashion, and there is one heterogeneous parameter ( ) associated with each neuron. Assuming that , for ω we consider an i.i.d. uniform or normal random variable of zero mean value; however, the results presented below are not restricted to these particular choices of the heterogeneity distribution. Note that the normal distribution needs to be truncated so that (hence ) is retained.
We choose the synaptic time τ to be heterogeneous, because overall clustering dynamics is sensitive to its variation. The presence of the heterogeneity in other parameters (such as g, , or I, as opposed to τ) would alter the detailed clustering dynamics in different ways (see Sect. 3 for details); however, whenever the strong correlation between the heterogeneity parameter and the variables develops in one way or another (see Sect. 3 for further details), the basic underpinning concept for the equation-free coarse-grained analysis presented in this paper can be again applied after an appropriate modification. In reality, all parameters are likely to be heterogeneous. The analysis of a network with multiple heterogeneities of this form is an interesting challenge, which is beyond the scope of the current study. In what follows, we consider the case , unless specified otherwise, and the width of the ω distribution (i.e., the standard deviation of ω, denoted by ) remains small compared to 1 so that the oscillators can still synchronize.
3 Cluster States
Given the parameter values presented in the previous section, an isolated neuron ( and ) undergoes a subcritical Andronov–Hopf bifurcation from a steady state as the external current reaches the value . The unstable periodic orbit born at this point eventually gains stability through a fold bifurcation of periodic orbits at . When two neurons are synaptically coupled together, the above-mentioned periodic orbits remain nearly unchanged, but the network exhibits bistability around the fold bifurcation point; the formation of clusters and the bistability itself has been well known . Our study is focused on the parameter regime corresponding to this two-neuron bistability, but it addresses the case of many neurons, heterogeneously coupled together (which exhibit further multi-stability).
The splitting into two clusters can occur in a number of different ways (i.e., different permutations) at the individual neuron level, resulting in various distinguishable double-cluster states. For a given realization of ω, depending on the initial configuration, each cluster in a double-cluster state consists of different neurons; in other words, the value of does not completely specify which cluster the i th neuron will join. Repeated simulations of various numbers of neurons with different initial configurations and/or independent random draws of ω, suggest that the sub-network sizes of the final stable double-cluster states tend to be almost the same ( neurons in one cluster and in the other, where ϵ is a small integer satisfying ). We note that an apparently different type of, or an extreme permutation of, stable double-cluster state, where the neurons split at the “middle” value of , is still possible. This is indeed a legitimate permutation; however, such a state is—we believe—highly unlikely to occur spontaneously.
4 Background: Detailed Dynamics of a Few Neurons
The complete dynamical analysis of even a relatively small number of, say ten, neurons is highly complicated because the number of possible clustering states rapidly increases with (scaling with the possible identity permutations). We obtain some initial insight into the generic dynamical features by analyzing small networks of neurons in detail.
In the case of four neurons with homogeneous coupling (i.e., identical ), the above-mentioned three PD points ( through in Fig. 5) collapse to a single point and the network exhibits a degenerate period-doubling bifurcation. Three Floquet multipliers simultaneously cross the unit circle at −1. All the cluster states of the same population size ratio then become indistinguishable, and for instance, the long term dynamics of single (double) cluster state is not different from that of a single (two) neuron(s).
We now compare our results with the predictions by Aronson et al., regarding the equivariant system of a population with homogeneous coupling . In the presence of symmetry, the equivariant branching lemma  leads to PD bifurcations on the single-cluster branch. The branches bifurcating off of the single cluster state branch are predicted to be of the form (corresponding to in our notation) with , where p and q are positive numbers representing the number of neurons in each cluster. Those authors showed that the double-cluster states may be stable for , and that the exact results for stability depend on the coefficients in the normal form at the bifurcations, which we do not attempt to obtain for the HH neuron model here. Other branches associated with other isotropy subgroups, the so-called “support solutions”, such as with for non-zero p, q and r, are predicted to emanate from the double-cluster state branches. Translating this prediction to the network of ten HH neurons, the or state is predicted to branch off from the single-cluster state, but and states may not directly branch off from there; rather, they may form as a consequence of subsequent bifurcations from the or double-cluster state. Our observations of the network with heterogeneous coupling are overall consistent with the above-mentioned predictions by Aronson et al. One non-trivial difference is that in our heterogeneously coupled HH neuron case, even three-cluster states with and groupings may branch directly off from the single-cluster branch (see Figs. 5 and 7).
5 A Coarse-Grained Description
5.1 Background: The Polynomial Chaos Expansion
where is the n-dimensional model variable, is the stochastic variable or parameter, an m-dimensional prescribed i.i.d. random variable each of which is drawn from the probability space . Here Ω is the sampling space, ℱ the σ field expanded by subsets of Ω, and μ the probability measure defined on ℱ. More complicated cases, e.g., where the dynamics is described by PDEs, can be formulated as well, however, such cases are not relevant to the current study.
In practice, the above expansion gets truncated at a certain order. Previous studies [35, 36] confirm that the orthonormal polynomials chosen from the Askey scheme for a given probability measure μ make the PC expansion converge exponentially with the rate of , where κ is a constant. However, the number of PC coefficients may rapidly increase as the random variable dimension m becomes larger, posing a computational challenge.
5.2 Coarse-Graining of the Clustering Dynamics
A computational dynamical analysis at the individual neuron level, such as the one presented in the previous section, is too complicated to perform for any realistic population size; a coarse-grained, population-level dynamical description and analysis become not only preferred, but necessary. Instead of keeping track of the state of every single neuron, we need to keep only a few collective descriptors of these states; yet, since the neurons are not homogeneous in their synaptic dynamics, a few moments of the distribution of the states are not sufficient: We need to not only know what the average and standard deviation of the states are, we also need to know which neurons (e.g. the low-τ or the high-τ ones) have low or high state values. In this joint distribution of neuron identities and neuron states, the marginal distribution of neuron states is not informative enough. That is why we turn to PC coefficients quantifying the correlation between the neuron identities and the neuron states. As was observed in the single cluster formation in a few different networks of oscillators [24, 27–29], a similar type of correlation between the dynamical variables ( , , , ) of the i th oscillator and its heterogeneity parameter rapidly develops in each of the clusters separately, during the initial transient (Fig. 2). The PC approach introduced to study the single cluster states  thus needs to be extended for the coarse-grained description of the double- and multiple-cluster states. In order to examine the possibility of applying the PC expansion to the double-cluster states, we first need to identify the distribution characteristics of the random (i.e., heterogeneity) parameters for each cluster, after the split.
When the network breaks up into two sub-networks, the original random parameters, ’s, are divided into two sets in a number of seemingly random ways, depending on the initial conditions of the neurons. Repeated numerical simulations from random initial configurations reveal that the random parameters for each cluster consistently span more or less the same range as the original random parameters (Fig. 2), and that the breaking of the original random parameter set into two subsets occurs in various permutations of the neuron identities. We quantitatively examine the statistical characteristics of the divided random parameters subsets using the Kolmogorov–Smirnov (KS) and the Wilk–Shapiro (WS) statistical tests , which compare the properties of an observed sample with those of the known distribution. As an illustrative example, we consider the case of a normal heterogeneity distribution.
where is a variable associated with the k th neuron belonging to the i th cluster, which consists of neurons. The first two coefficients have the following geometrical meaning on the coarse-grained level: is the average value, and measures the level of linear spread of the variable among the neurons around the average value , as a consequence of the heterogeneity. For the case of the membrane potential (when is ), measures the average potential, and roughly measures the instantaneous spread of the potential among the neurons in the i th cluster. The higher order PC coefficients are related to higher order moments of the spread of the individual neuron’s variables in each cluster.
The individual-level details, such as the exact composition of the neurons in each cluster, vary among different initial conditions and different draws of the random variable ω. However, the temporal trajectories of the PC coefficients remain robust over such microscopically distinguishable states, with a small level of statistical fluctuation. The PC expansion Eq. (11) converges rapidly; the magnitudes of rapidly decrease with increasing j (Fig. 9), as expected from the Askey scheme. Upon ensemble averaging, the PC description provides an appropriate statistical representation of the coarse-grained state.
So far, the random parameters in the divided clusters are assumed to be described by the same distribution as the original one for the entire network based on the findings of the statistical tests. However, even if statistically unlikely, the previously mentioned extreme case of “split-in-the-middle” state where one cluster is formed by the neurons of ( is a specific value around the middle value of 0) while the other cluster consists of neurons with the remaining values of ω, does exist; an artificially prepared double cluster state conforming to this grouping (whether or not) is indeed found to be stable. There exist only few limit cycle solutions of this type, and such states would be statistically insignificant in the coarse-grained description. Should such a split arise, the heterogeneity characteristics of each sub-network is clearly inconsistent with the full heterogeneity distribution. In this case, the heterogeneity sub-domain corresponding to each cluster should be treated separately to account for the split at . A variant or extension of the multi-element PC method developed for stochastic differential equations  should be considered in this case.
6 Coarse-Grained Computations
In this section, we perform equation-free coarse-grained computations for double cluster states, treating each of them separately. By doing this, we circumvent the steps deriving the model equations for the PC coefficients (Eq. (9)) for the current system. We do not identify the coarse-grained model equations; however, we analyze the dynamics by computationally obtaining the solutions to those equations. This approach does not rely on any simplifying assumptions, such as weak coupling, as long as synchronization occurs. It is not limited to a particular choice of the distribution for the random parameters and, in principle, it works equally well both for “large” finite and infinite network sizes. The success of this method attests to the accurate and sufficient description of the network by the chosen few coarse-grained variables.
In order for a coarse-grained calculation of double-clusters to be feasible, the neurons belonging to different clusters need to be systematically identified and grouped together. This can be done in the following way: As the variation of the dynamical variables within a cluster is much less than that between two clusters most of the time during a cycle (Fig. 2), the neurons belonging to different clusters can be differentiated by measuring the temporal correlation of their dynamical variables. The time series of the neuron variables observed over a length of time (still a fraction of the period) is sampled at a set of intervals, say at every time interval of ; then the correlation of the sampled time series is calculated. A threshold is applied to the correlation matrix; matrix entries are set to 1 if above the threshold and 0 otherwise. The thresholded correlation matrix can be interpreted as the adjacency matrix of the network of neurons. The first non-trivial eigenvector of the adjacency matrix reveals clustering of the neurons. The entries of this eigenvector are clearly clustered around two distinct values. Projections of the eigenvector onto the different neurons are sorted by these values, thereby identifying two clusters.
We start the equation-free coarse-grained analysis by integrating the double-cluster states in time, using a forward Euler coarse projective integration method  (which does not differ conceptually from the coarse integration of single-cluster states). The first three PC coefficients for each dynamical variable are retained as the coarse-grained variables (truncating at is enough for general purposes, per the convergence results seen in Fig. 9). This method, in which a forward Euler—or other choice—projection algorithm is directly applied to the time evolution of the coarse-grained variables, is the simplest demonstration of the applicability of equation-free computations. Each iteration in this algorithm consists of a few steps; healing, microscopic evolution (direct integration of the full model), and the projection of the coarse-grained variables. The number of time steps in each of the healing, direct integration, and jump steps can be fixed for the entire time evolution or be adaptively changed for better efficiency .
Both of the demonstrated equation-free algorithms successfully compute the correct dynamical states, confirming that a few PC expansion coefficients are appropriate coarse-grained dynamical variables, enabling the description and the dynamical analysis of the large ensemble of neurons at that level.
Any “system of systems” in practice, including a network of neurons studied here, is unlikely to consist of homogeneously coupled identical entities, and the consideration of heterogeneity among the constituent entities is often necessary. The heterogeneity may introduce fundamental differences into the dynamics, compared to the homogeneous case. The oscillating entities in each cluster are now no longer completely synchronized, the dynamical dimension of the network increases tremendously, and the individual-level dynamics and the corresponding dynamical analysis in a traditional way could be much more complicated. Even though some of the qualitative features may remain the same as in the homogeneous case, the detailed dynamics often cannot easily be deduced from that of the homogeneous limit. Furthermore, the system size is often finite, and an analysis treating it as an infinitely large system, where the heterogeneity parameter is assumed to be continuous, might not be appropriate. The equation-free coarse-grained computational method presented here is well suited to such cases, without requiring any of simplifying assumptions.
In a heterogeneously coupled network of Hodgkin–Huxley neurons, as studied here, all the combinatorially different ways of transitioning between the single- and the double-cluster states are distinguishable at the individual neuron level. The composition of the neurons in each of the double-cluster states is apparently “randomly” decided, depending on the initial configuration. We see that the random variables in each sub-network are statistically not inconsistent with those of the original, entire network. This enables each of the double clusters to be described and analyzed in the equation-free polynomial chaos framework that has already successfully been applied to single-cluster states in several other examples. Our approach, based on the strong correlations among the variables which rapidly develop in each of the clusters separately during the initial transient, gives rise to a low-dimensional description of a large heterogeneous network. The approach is applicable to a range of oscillator models exhibiting the same type of splitting of random parameters in the formation of double clusters. Though this work focuses on neurons splitting into two groups (in networks of Hodgkin–Huxley neurons), the techniques used here constitute a first step that can be extended for different types of oscillators and different number of groups or clusters, as long as the correlation remains valid. For instance, in the case of slightly different types of double clusters (when I is heterogeneous; refer to Fig. 4), our method should be extended, in line with some variant of the multi-element PC method developed for stochastic differential equations .
The authors would like to thank M. Krupa for his helpful comments. The work of I.G.K. and K.R. was partially supported by the US Department of Energy, while the work of C.R.L. was supported by the Marsden Fund Council, administered by the Royal Society of New Zealand. The work of J.C. was supported by Fondecyt grant 1140143.
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