^{1}, Kristina Todorović

^{2}, Nebojša Vasović

^{3}and Nikola Burić

^{4,a)}

### Abstract

Properties of spontaneously formed clusters of synchronous dynamics in a structureless network of noisy excitable neurons connected via delayed diffusive couplings are studied in detail. Several tools have been applied to characterize the synchronizationclusters and to study their dependence on the neuronal and the synaptic parameters. Qualitative explanation of the cluster formation is discussed. The interplay between the noise, the interaction time-delay and the excitable character of the neuronal dynamics is shown to be necessary and sufficient for the occurrence of the synchronizationclusters. We have found the two-cluster partitions where neurons are firmly bound to their subsets, as well as the three-cluster ones, which are dynamical by nature. The former turn out to be stable under small disparity of the intrinsic neuronal parameters and the heterogeneity in the synaptic connectivity patterns.

Synchronization of neuronal activity is considered as a fundamental dynamical ingredient of various brain functions. Therefore, multifarious properties of the synchronizationdynamics in neural network models have been the topic of intensive and thorough research. Dynamics of realistic neuronal networks is crucially influenced by the interaction involving synaptic delays, and by small perturbations of various origin, which are commonly treated as noise. The impacts that noise or the interaction delays make on the dynamical properties of neuronal populations are diverse and well known. However, the systematic exploration of the co-effects introduced by the simultaneous presence of noise and time delay has begun only quite recently. An interesting type of the ensuing synchronization phenomena may be the splitting of a population into clusters, each made up of neurons oscillating in synchrony, whereby the activity between the clusters shows phase lags. Usually, the synchronizationcluster formation is a consequence of some structural inhomogeneity, either in the local neuronal parameters or the network topology. In a recent letter, we briefly reported on the occurrence of synchronizationclusters observed in a homogeneous network of excitable stochastic neurons with delayed coupling. Despite the network being completely structureless, the interplay of noise, time-delay and the excitable character of the neurons is found sufficient to trigger the clustering. This paper provides a thorough and systematic exploration of such spontaneous cluster formation.

This work was supported in part by the Ministry of Education and Science of the Republic of Serbia, under Project Nos. 171017 and 171015.

I. INTRODUCTION

II. BACKGROUND ON THE NEURON MODEL AND THE POPULATION DYNAMICS

III. OBSERVATION OF CLUSTERING

A. Where to look for the cluster states?

IV. PROPERTIES OF THE TWO-CLUSTER STATE DYNAMICS

A. Asymptotic dynamics

V. THREE-CLUSTER STATE DYNAMICS

A. Dynamical correlation coefficients as means to quantify dynamical clustering

VI. EXPLANATION OF THE CLUSTERINGDYNAMICS

VII. THE MF MODEL AND CLUSTERING

VIII. SUMMARY AND DISCUSSION

### Key Topics

- Cluster analysis
- 47.0
- Cluster dynamics
- 33.0
- Nerve cells
- 29.0
- Coherence
- 22.0
- Networks
- 14.0

## Figures

Recap on the dynamics of an excitable Fitzhugh-Nagumo element, followed by an overview on the collective modes of excitable media, illustrated for the system 1 at . (a) In case of CR, the stochastic LC is a precursor to the deterministic one. As such, it takes place on the attractive (outer) branches of the *x* nullcline (dotted line), avoiding the unstable (middle) branch. *EQ* indicates the position of the equilibrium. A typical orbit (solid line) is made up of two portions of slow motion *O*(1), connected by two rapid transients . The former include a descent down the refractory branch until the left knee is reached, and the ascent along the spiking branch . The inset shows a section from the time series *x*(*t*) for *D* = 0.003. (b) For instantaneous couplings, the neural population exhibits three generic types of global behavior if *D* is systematically increased. This is epitomized by the phase portraits of the ensemble averages and at *c* = 0.1, displaying incoherent motion (*D* = 0.0002), coherent collective oscillations (*D* = 0.002) and the decay into the chaotic regime (*D* = 0.009).

Recap on the dynamics of an excitable Fitzhugh-Nagumo element, followed by an overview on the collective modes of excitable media, illustrated for the system 1 at . (a) In case of CR, the stochastic LC is a precursor to the deterministic one. As such, it takes place on the attractive (outer) branches of the *x* nullcline (dotted line), avoiding the unstable (middle) branch. *EQ* indicates the position of the equilibrium. A typical orbit (solid line) is made up of two portions of slow motion *O*(1), connected by two rapid transients . The former include a descent down the refractory branch until the left knee is reached, and the ascent along the spiking branch . The inset shows a section from the time series *x*(*t*) for *D* = 0.003. (b) For instantaneous couplings, the neural population exhibits three generic types of global behavior if *D* is systematically increased. This is epitomized by the phase portraits of the ensemble averages and at *c* = 0.1, displaying incoherent motion (*D* = 0.0002), coherent collective oscillations (*D* = 0.002) and the decay into the chaotic regime (*D* = 0.009).

Characterization of the cluster states in terms of features of the corresponding binary and weighted coherence networks. In (a) and (b), it is demonstrated how the distinction between the homogeneous coherent states and the *n*-cluster states can be made explicit by the binary coherence network, which possesses a unimodal (an *n*-modal) distribution of the nodal connectedness degrees *P*(*k*) in the former (latter) case. The data in (a) are obtained for the homogeneous coherent state at *c* = 0.1, *D* = 0.001, , whereas the parameter values for the two-cluster state in (b) are *c* = 0.1, *D* = 0.00025, . The weighted coherence network, represented by the weight matrix in (c), and the binary network in (d) may serve independently or combined to capture the structure of the given cluster state, as shown for the two-fraction partition at *c* = 0.1, *D* = 0.0005, .

Characterization of the cluster states in terms of features of the corresponding binary and weighted coherence networks. In (a) and (b), it is demonstrated how the distinction between the homogeneous coherent states and the *n*-cluster states can be made explicit by the binary coherence network, which possesses a unimodal (an *n*-modal) distribution of the nodal connectedness degrees *P*(*k*) in the former (latter) case. The data in (a) are obtained for the homogeneous coherent state at *c* = 0.1, *D* = 0.001, , whereas the parameter values for the two-cluster state in (b) are *c* = 0.1, *D* = 0.00025, . The weighted coherence network, represented by the weight matrix in (c), and the binary network in (d) may serve independently or combined to capture the structure of the given cluster state, as shown for the two-fraction partition at *c* = 0.1, *D* = 0.0005, .

Focus on the impact of interaction delay on the system's behavior. (a) shows the plots for *D* = 0.0005 (solid circles) and *D* = 0.0007 (open diamonds) at fixed *c* = 0.1. The local minima exhibited by serve as an indication on the intervals of that foster the cluster states. (b) The representation scheme with respect to *D* is adopted from (a). The inset refers to the variation of the average oscillation period with for the macroscopic variable *X*. In qualitative terms, the curves appear virtually the same for different noise, whereby the given profiles imply that the system's dynamics can be traced to the competition between the noise-driven and the delay-driven oscillation modes. The dashed line corresponds to the case . In the main frame is displayed the dependence of the scaled average oscillation period on for different *D*. For the homogeneous coherent states found within the approximate intervals and , the above competition is resolved in favor of the delay-driven mode. For the two-cluster states around and , the noise-driven mode prevails.

Focus on the impact of interaction delay on the system's behavior. (a) shows the plots for *D* = 0.0005 (solid circles) and *D* = 0.0007 (open diamonds) at fixed *c* = 0.1. The local minima exhibited by serve as an indication on the intervals of that foster the cluster states. (b) The representation scheme with respect to *D* is adopted from (a). The inset refers to the variation of the average oscillation period with for the macroscopic variable *X*. In qualitative terms, the curves appear virtually the same for different noise, whereby the given profiles imply that the system's dynamics can be traced to the competition between the noise-driven and the delay-driven oscillation modes. The dashed line corresponds to the case . In the main frame is displayed the dependence of the scaled average oscillation period on for different *D*. For the homogeneous coherent states found within the approximate intervals and , the above competition is resolved in favor of the delay-driven mode. For the two-cluster states around and , the noise-driven mode prevails.

Insight on the impact of *D* and *c* on the system's dynamics. (a) shows the family of curves for the set of delay values including (open triangles), (solid triangles), as well as (open circles), having *c* = 0.1 fixed. Two-cluster states are indicated for the noise amplitudes at , whereas the appropriate range of *D* is broader at . The curve's profile for implies the lack of clustering within the considered interval of noise. (b) illustrates how the shape of the curves at fixed is altered under variation of *c*, beginning with *c* = 0.08 (open triangles), over *c* = 0.1 (solid triangles) to *c* = 0.12 (open circles). Too strong a coupling appears to suppress the onset of the cluster states.

Insight on the impact of *D* and *c* on the system's dynamics. (a) shows the family of curves for the set of delay values including (open triangles), (solid triangles), as well as (open circles), having *c* = 0.1 fixed. Two-cluster states are indicated for the noise amplitudes at , whereas the appropriate range of *D* is broader at . The curve's profile for implies the lack of clustering within the considered interval of noise. (b) illustrates how the shape of the curves at fixed is altered under variation of *c*, beginning with *c* = 0.08 (open triangles), over *c* = 0.1 (solid triangles) to *c* = 0.12 (open circles). Too strong a coupling appears to suppress the onset of the cluster states.

Two-cluster states at small *D* and . (a) As a signature of the population split, the phase portrait for the collective dynamics shows a twisted limit cycle orbit, where the two discernible segments reflect the action of the clusters. (b) A section from the *X*(*t*) series (dashed line) is overlaid by the series (solid lines) for two arbitrary neurons from the distinct clusters. A high-level coherence within the subsets is witnessed by the fact that the peaks of the global potential perfectly match the ones of the local potentials. The latter series imply that the firing of clusters is locked in antiphase. The data are provided for the case *c* = 0.1, *D* = 0.00025, .

Two-cluster states at small *D* and . (a) As a signature of the population split, the phase portrait for the collective dynamics shows a twisted limit cycle orbit, where the two discernible segments reflect the action of the clusters. (b) A section from the *X*(*t*) series (dashed line) is overlaid by the series (solid lines) for two arbitrary neurons from the distinct clusters. A high-level coherence within the subsets is witnessed by the fact that the peaks of the global potential perfectly match the ones of the local potentials. The latter series imply that the firing of clusters is locked in antiphase. The data are provided for the case *c* = 0.1, *D* = 0.00025, .

Properties of single neuron dynamics. (a) The distribution of local jitters implies that highly regular spiking patterns are maintained across the ensemble. (b) The first return map of the firing periods for an arbitrary neuron illustrates how any larger deviations from the mean value are rare, further subdued already within the following cycle. The latter is upheld independent on the particular cluster a neuron belongs to. This is witnessed in the inset, which shows the ISI distributions for two arbitrary neurons from the distinct subsets. The parameter set is .

Properties of single neuron dynamics. (a) The distribution of local jitters implies that highly regular spiking patterns are maintained across the ensemble. (b) The first return map of the firing periods for an arbitrary neuron illustrates how any larger deviations from the mean value are rare, further subdued already within the following cycle. The latter is upheld independent on the particular cluster a neuron belongs to. This is witnessed in the inset, which shows the ISI distributions for two arbitrary neurons from the distinct subsets. The parameter set is .

Two-cluster states at intermediate *D* and . Phase portrait for the global dynamics at is projected in the *X* – *Y* plane. The properties of the cluster partition are seen to depend on the parameter values, whereby the larger *D* and appear to favor the states closer to an equipartition over the asymmetric clustering.

Two-cluster states at intermediate *D* and . Phase portrait for the global dynamics at is projected in the *X* – *Y* plane. The properties of the cluster partition are seen to depend on the parameter values, whereby the larger *D* and appear to favor the states closer to an equipartition over the asymmetric clustering.

In case of the two-cluster state, evolution of the representative clouds for the distinct clusters is shown in the phase space. To indicate the stationary character of the cluster partition, we selected a triplet of neurons, including two arbitrary members (labeled by 1 and 162) from one cluster, and a single neuron (labeled by 51) from the other cluster. At any given moment, the neurons' respective positions, denoted by arrows, imply that there is no mixing between the clusters. The parameters are set to .

In case of the two-cluster state, evolution of the representative clouds for the distinct clusters is shown in the phase space. To indicate the stationary character of the cluster partition, we selected a triplet of neurons, including two arbitrary members (labeled by 1 and 162) from one cluster, and a single neuron (labeled by 51) from the other cluster. At any given moment, the neurons' respective positions, denoted by arrows, imply that there is no mixing between the clusters. The parameters are set to .

dependence for the two-cluster state at . The existence of an asymptotic component suggests the persistence of clustering in the thermodynamic limit, whereby the onset ofthe near-asymptotic behavior is found about . The latter makes it unlikely that the stability of the two-cluster states in larger populations may be altered by some mechanisms absent at smaller *N*.

dependence for the two-cluster state at . The existence of an asymptotic component suggests the persistence of clustering in the thermodynamic limit, whereby the onset ofthe near-asymptotic behavior is found about . The latter makes it unlikely that the stability of the two-cluster states in larger populations may be altered by some mechanisms absent at smaller *N*.

Global properties of the three-cluster state. (a) Compared to Fig. 2(c), the weight matrix for the weighted coherence network displays larger off-diagonal terms, indicating less clear cluster separation. (b) The binary coherence network reveals the three-cluster partition if the threshold level is raised to . The data refer to the parameter set .

Global properties of the three-cluster state. (a) Compared to Fig. 2(c), the weight matrix for the weighted coherence network displays larger off-diagonal terms, indicating less clear cluster separation. (b) The binary coherence network reveals the three-cluster partition if the threshold level is raised to . The data refer to the parameter set .

Illustration of the dynamical clustering typifying the three-cluster states. The top and bottom panels show sections from the series for two arbitrary neurons, labeled 2 and 82, which belong to distinct core-clusters. The middle panel refers to a minority subset that exhibits switching between the cores, with its behavior characterized by the neuron 40. Within the interval , spiking in the middle series is in step with the top series, whereas for it is synchronized with the firing series from the bottom panel. The moment when the neuron 40 jumps between the two core-clusters is indicated by an arrow in the middle panel. The data are provided for .

Illustration of the dynamical clustering typifying the three-cluster states. The top and bottom panels show sections from the series for two arbitrary neurons, labeled 2 and 82, which belong to distinct core-clusters. The middle panel refers to a minority subset that exhibits switching between the cores, with its behavior characterized by the neuron 40. Within the interval , spiking in the middle series is in step with the top series, whereas for it is synchronized with the firing series from the bottom panel. The moment when the neuron 40 jumps between the two core-clusters is indicated by an arrow in the middle panel. The data are provided for .

Asymptotic vs intermittent synchronization between the neurons. (a) and (b) refer to the two-cluster state at and the three-cluster state for , respectively. In both panels are plotted the time variations of for an arbitrary pair of neurons in the same cluster (solid lines) and from the distinct clusters (dotted lines). (a) implies stable correlated (anti-correlated) spiking within (between) the subsets. The mixed picture in (b) indicates that correlated episodes occur for neurons occupying both the same cluster and the distinct ones, but are more abundant in the former case. The smoothed curves are obtained by applying the second-order Savitzky-Golay algorithm.

Asymptotic vs intermittent synchronization between the neurons. (a) and (b) refer to the two-cluster state at and the three-cluster state for , respectively. In both panels are plotted the time variations of for an arbitrary pair of neurons in the same cluster (solid lines) and from the distinct clusters (dotted lines). (a) implies stable correlated (anti-correlated) spiking within (between) the subsets. The mixed picture in (b) indicates that correlated episodes occur for neurons occupying both the same cluster and the distinct ones, but are more abundant in the former case. The smoothed curves are obtained by applying the second-order Savitzky-Golay algorithm.

Analysis of the local dynamics in analogy to motion of particles in a double-well potential. The solid and dotted curves indicate the respective potentials attributed to representative neurons from the distinct clusters. Transitions between the refractory and the spiking branches are interpreted as hopping over the potential barrier, whose height depends on the interaction term. (a) reflects the setup where one neuron is active, and the other is refractory. The configuration in (b) shows both neurons on the refractory branch, with one having just completed a spike, whereas the other approaches the left knee. In (c) one of the neurons is trapped on the refractory branch, which constitutes a hallmark of clustering on the local level.

Analysis of the local dynamics in analogy to motion of particles in a double-well potential. The solid and dotted curves indicate the respective potentials attributed to representative neurons from the distinct clusters. Transitions between the refractory and the spiking branches are interpreted as hopping over the potential barrier, whose height depends on the interaction term. (a) reflects the setup where one neuron is active, and the other is refractory. The configuration in (b) shows both neurons on the refractory branch, with one having just completed a spike, whereas the other approaches the left knee. In (c) one of the neurons is trapped on the refractory branch, which constitutes a hallmark of clustering on the local level.

(a) and (b) display the phase portraits for the local dynamics of the homogeneous coherent states and the cluster states, respectively. The latter are distinguished by the kink *K*, which reflects the co-effect. The inset shows sections of series for two arbitrary members of the distinct clusters, whereby an arrow indicates the neuron, whose portrait is presented in the main frame. For an insight into the role of kink, we highlighted by bullets the respective positions of the involved neurons along the orbit at a given moment *t*. As a neuron lies at the kink, the other's potential is at the peak of the rising phase(farthest right bullet). In addition, at , the former is located very close to the peak, which illustrates how the delay affects the adjustment between the clusters' dynamics. The parameters are set to in (a) and in (b).

(a) and (b) display the phase portraits for the local dynamics of the homogeneous coherent states and the cluster states, respectively. The latter are distinguished by the kink *K*, which reflects the co-effect. The inset shows sections of series for two arbitrary members of the distinct clusters, whereby an arrow indicates the neuron, whose portrait is presented in the main frame. For an insight into the role of kink, we highlighted by bullets the respective positions of the involved neurons along the orbit at a given moment *t*. As a neuron lies at the kink, the other's potential is at the peak of the rising phase(farthest right bullet). In addition, at , the former is located very close to the peak, which illustrates how the delay affects the adjustment between the clusters' dynamics. The parameters are set to in (a) and in (b).

Sequence of Hopf bifurcation curves for the MF model under increasing *D* and . Though the curves can account for the transitions between the stochastically stable fixed point and the stable limit cycle, one cannot associate them with the cluster formation.

Sequence of Hopf bifurcation curves for the MF model under increasing *D* and . Though the curves can account for the transitions between the stochastically stable fixed point and the stable limit cycle, one cannot associate them with the cluster formation.

Behavior of the MF model in the parameter domains related to clustering. (a) and (b) illustrate bistability observed for *D* = 0.00025, at *c* = 0.1. (a) Shows the examples of trajectories converging either to the fixed point or the limit cycle, contingent on the initial conditions. In (b) are mapped the corresponding basins of attraction, with the equilibrium (*EQ*) found to lie very close to their boundary. (c) Refers to the “fortunate failure” of the approximate model under the increased *D* and . The time series and the phase portrait are provided for .

Behavior of the MF model in the parameter domains related to clustering. (a) and (b) illustrate bistability observed for *D* = 0.00025, at *c* = 0.1. (a) Shows the examples of trajectories converging either to the fixed point or the limit cycle, contingent on the initial conditions. In (b) are mapped the corresponding basins of attraction, with the equilibrium (*EQ*) found to lie very close to their boundary. (c) Refers to the “fortunate failure” of the approximate model under the increased *D* and . The time series and the phase portrait are provided for .

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