Coarse-Grained Clustering Dynamics of Heterogeneously Coupled Neurons

A network of oscillators can form sets of sub-networks oscillating with phase lags among them [1]; 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 [5], in networks of inhibitory reticular thalamic nucleus (RTN) neurons [6], in phase oscillators with nearest neighbor coupling [7, 8], in models for synthetic genetic networks [9], and for identical Hodgkin–Huxley (HH) neurons with homogeneous, weak coupling [10], 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 [5]. 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 [10]. A generalization of the analytical framework for the Kuramoto-like coupled phase oscillators, including higher Fourier modes, has been attempted [13]. 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 [24]. The approach is based on the Polynomial Chaos (PC), also known as Wiener’s chaos expansion [25], originally introduced to model stochastic processes with Gaussian random variables using Hermite polynomials; it has been further developed and widely used for uncertainty quantification [26]. 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 [27], van der Pol oscillators [28], and simplified neuron models [29].

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 study ensembles of Hodgkin–Huxley neurons. The dynamical state of each of the HH neurons is described by a set of variables , where V is the membrane potential, m and h are the activation and inactivation variables of the sodium current, and n is the activation variable of the potassium current. The equations for the i th neuron read [30]

(1)

where I is the external current (an important control parameter in our study), is the synaptic current entering the i th neuron (see Eq. (2) below), and

for , and n, where

We use almost the same parameter values as in [10], 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 synaptic current for each neuron in a network of N all-to-all coupled neurons is modeled as

(2)

where , g is the coupling strength among the neurons (which is mostly set to 3.0 in this paper), and the synaptic variable is governed by

(3)

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.

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 [10]. 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).

In this regime, the network can realize several types of stable periodic behavior: The entire network may oscillate synchronously (Fig. 1(a); single-cluster state), or may break up into two or more sub-networks (or clusters), each of them synchronously oscillating with a phase lag between the clusters (Fig. 1(b); double-cluster state). In each cluster, the trajectories of the constituting neurons get slightly “dispersed” as a consequence of the heterogeneity. The period of the double-cluster state is almost twice of that of the single cluster state (comparing Fig. 1(a) and (b)), and the transition between the single- and double-cluster states is related to the two-neuron period-doubling (PD) bifurcation (refer to Sect. 4 for further details). Such cluster states are observed for any network size.

The projection of a double-cluster state onto a certain phase plane of dynamical variables (right panels of Fig. 2) sheds light on an important aspect of the clustering dynamics; it reveals that (i) a strong correlation between the heterogeneity parameter ω and the dynamical variables separately develops for each cluster and (ii) that the neurons break up almost evenly in ω to form two clusters. The same type of correlation develops for the entire population of single-cluster states in a few different coupled oscillator models, which have been studied with a PC coarse-grained description and a subsequent equation-free analysis [24, 27–29]. Here, we attempt to extend this framework to apply to multiple-cluster states as well. Similar single- and double-cluster states exist also in homogeneous networks of identical HH neurons [10]; in the absence of heterogeneity, however, the constituting neurons of each cluster get synchronized completely, which naturally reduces the population-level dynamical dimension. In that case, each of the clusters can be treated as a fictitious neuron of appropriate weight or rescaling factor being assigned [31, 32], and the overall dynamics is effectively the same as that of a few/several neurons.

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.

We consider the cases where the heterogeneity exists in other model parameters as well. For instance, when g is given as , where and ω is a uniform random variable, while τ is given as a fixed value of 1.0 throughout the neurons, we see almost the same behavior as in the case of a heterogeneous time constant presented above (Fig. 3). In case I is heterogeneous, in addition to the same type of single-cluster states, a slightly different type of double-cluster states are observed (Fig. 4); while the “identity-state” correlation still exists, only a specific range of ω breaks up to form double-cluster states. In this case, the coarse-graining method discussed in this study will have to be further extended; we will leave this as a topic for future study.

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.

We start with four neurons, where ’s are distributed evenly in , i.e., ; at this population size, the characterization of the distribution function does not have statistical significance. The detailed bifurcation diagrams in this section are obtained using AUTO [33]. When the stable single-cluster (period-1) solution branch is continued in the decreasing direction of I, it loses the stability through a Period Doubling (PD) bifurcation, which is denoted by in Fig. 5. As the current I decreases even further, there arise two more PD bifurcations ( and in Fig. 5). Continuation of the bifurcated branch emanating from the first PD point exhibits unstable period-2 double-cluster states, consisting of two clusters having two neurons in each cluster (referred to as a split, where the notation means m neurons belong to one cluster and the remaining n ( ) neurons belong to the other). After a sequence of fold bifurcations (saddle-nodes of limit cycles, filled circles in Fig. 5), this unstable period-2 branch eventually becomes stable, giving rise to a stable double-cluster state of split with a grouping of , i.e., neurons 1 and 4 form one cluster synchronously oscillating together, while neurons 2 and 3 form the other. For convenience, the neurons are labeled from 1 to N in the increasing order of the value for . Continuation of the solution branches bifurcated from and results in unstable three-cluster states with a split, i.e., states for which two neurons oscillate in synchrony and the remaining two “clusters” contain one neuron each. These two branches also undergo numerous fold bifurcations, but they never gain stability.

Starting from a stable state, we obtain similar double-cluster states by swapping the dynamical variables among the neurons (while keeping ’s and all the other parameters unchanged) and directly integrating the model from this initial condition. We find that the solution branches for the other two combinations of evenly split double-cluster states of , and states, are stable over a finite parameter range, forming isolas (Fig. 6; the single cluster state branch in Fig. 5 is included for comparison). The isolas have both stable (solid line) and unstable (dashed line) segments separated by fold bifurcation points (filled circles).

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).

The full bifurcation diagram for larger populations contains many more branches and bifurcation points, which rapidly becomes too complicated (and probably pointless) to analyze. It would help to consider the clustering dynamics of a few different progressively increasing network sizes, which would allow us to induce the dynamics of a larger network. Our analysis of four and ten neurons (see below) indicates that the essential features appear to remain the same. A bifurcation diagram for ten heterogeneous neurons including some of the relevant branches discussed above, is again obtained using AUTO. The ’s are again uniformly distributed in for simplicity. The period-1 single-cluster state has nine PD bifurcations, through in Fig. 7; in general, there exist PD bifurcations on the single-cluster solution branch of a network of N neurons. As in smaller networks, continuing the branch bifurcating from leads to a double-cluster state with an equal-sized ( ) split which eventually gains stability through a fold bifurcation. The continuation of branches bifurcating from and gives rise to unstable three-cluster states, and , respectively, which never become stable. As was done with four neurons, we combinatorially swap the dynamical states of the neurons comprising a stable double-cluster, in order to obtain other stable double-cluster states. The original split state has the grouping of in the case of our random initial condition, where 0 represents the tenth neuron ( ). We obtained several other stable splits, including , , , and even a stable split, , all of which lie on solution branches forming isolas. These five stable branches are located very close to one another in phase and parameter space (Fig. 7). We expect that, in this version of “attractor crowding” [11, 12], small stochastic or deterministic perturbations may cause the dynamics to easily “flip” basins of attraction and approach nearby “coarsely indistinguishable” limit cycles. We checked other network sizes and found that, regardless of the population size, there exists a range of parameters for which multi-stability between single- and double-cluster states is observed, and the stable double-cluster states consist of nearly equal-sized clusters.

We now compare our results with the predictions by Aronson et al., regarding the equivariant system of a population with homogeneous coupling [5]. In the presence of symmetry, the equivariant branching lemma [34] 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).

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 [40] (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 [41].

We implement the coarse projective integration algorithm with fixed time-step sizes for each of the healing, evaluation, and jump steps, selected to accommodate accuracy and stability even when the variables rapidly change around a spike; these step sizes are almost certainly “overly cautious” during the slow recovery phase of the neuron. The small errors between coarse projective integration and direct full integration (which depend on the projection step size and the projection method) can be seen in Fig. 10. Projective integration for the cluster states needs to be implemented with caution because of the network’s multi-stability.

A more sophisticated algorithm can be used to compute the coarse-grained periodic solution of the double clusters, utilizing an equation-free fixed point algorithm [40] and the PC expansion. Standard Newton–Raphson-like fixed point algorithms require evaluation of both the coarse residual and the coarse Jacobian. The coarse-grained level equations are evaluated with the coarse time-stepper, and the evaluation of the coarse Jacobian is circumvented using a Krylov subspace method, which requires only evaluating the action of this Jacobian on specified vectors [42]. The coarse-grained representation of a typical solution is found this way. The individual and coarse solutions can deviate slightly, as only a finite number of PC coefficients are used during the computations, but the difference is practically unnoticeable in the “eye norm” (Fig. 11).

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 [39].