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Edited by: Ramon Guevara Erra, Laboratoire Psychologie de la Perception (CNRS), France

Reviewed by: Zoran Levnajić, Institute Jozef Stefan, Slovenia; Peter J. Thomas, Case Western Reserve University, USA; Hiroya Nakao, Tokyo Institute of Technology, Japan

*Correspondence: Rodrigo Echeveste

†Present Address: Rodrigo Echeveste, Computational and Biological Learning Lab, Cambridge University, Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, UK

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The study of balanced networks of excitatory and inhibitory neurons has led to several open questions. On the one hand it is yet unclear whether the asynchronous state observed in the brain is autonomously generated, or if it results from the interplay between external drivings and internal dynamics. It is also not known, which kind of network variabilities will lead to irregular spiking and which to synchronous firing states. Here we show how isolated networks of purely excitatory neurons generically show asynchronous firing whenever a minimal level of structural variability is present together with a refractory period. Our autonomous networks are composed of excitable units, in the form of leaky integrators spiking only in response to driving currents, remaining otherwise quiet. For a non-uniform network, composed exclusively of excitatory neurons, we find a rich repertoire of self-induced dynamical states. We show in particular that asynchronous drifting states may be stabilized in purely excitatory networks whenever a refractory period is present. Other states found are either fully synchronized or mixed, containing both drifting and synchronized components. The individual neurons considered are excitable and hence do not dispose of intrinsic natural firing frequencies. An effective network-wide distribution of natural frequencies is however generated autonomously through self-consistent feedback loops. The asynchronous drifting state is, additionally, amenable to an analytic solution. We find two types of asynchronous activity, with the individual neurons spiking regularly in the pure drifting state, albeit with a continuous distribution of firing frequencies. The activity of the drifting component, however, becomes irregular in the mixed state, due to the periodic driving of the synchronized component. We propose a new tool for the study of chaos in spiking neural networks, which consists of an analysis of the time series of pairs of consecutive interspike intervals. In this space, we show that a strange attractor with a fractal dimension of about 1.8 is formed in the mentioned mixed state.

The study of collective synchronization has attracted the attention of researchers across fields for now over half a century (Winfree,

Different degrees of collective synchronization may occur also in networks of elements emitting signals not continuously, such as limit-cycle oscillators, but via short-lived pulses (Mirollo and Strogatz,

The individual elements are usually modeled in this context as integrate and fire units (Kuramoto,

Here τ is the characteristic relaxation timescale of _{θ}, a pulse is emitted (the only information carried to other units) and the internal variable is reset to _{rest}.

These units are usually classified either as oscillators or as excitable units, depending on their intrinsic dynamics. The unit will fire periodically even in the absence of input when _{θ}). Units of this kind are denoted

A natural frequency given by the inverse integration time of the autonomous dynamics exist in the case of pulse-coupled oscillators. There is hence a preexisting, albeit discontinuous limit cycle, which is then perturbed by external inputs. One can hence use phase coupling methods to study networks of pulse coupled oscillators (Mirollo and Strogatz,

These assumptions break down for networks of coupled excitable units as the ones here described. In this case the units will remain silent without inputs from other elements of the system and there are no preexisting limit cycles and consequently also no preexisting natural frequencies (unlike

The study of pulse coupled excitable units is of particular relevance within the neurosciences, where neurons are often modeled as spike emitting units that continuously integrate the input they receive from other cells (Burkitt,

Alternatively, one could have built networks of excitatory neurons with high variability in the connection parameters, reproducing realistic connectivity distributions, such as those found in the brain. The large number of parameters involved would make it however difficult to fully characterize the system from a dynamical systems point of view, the approach taken here. An exhaustive phase-space study would also become intractable. We did hence restrict ourselves in the present work to a scenario of minimal variability, as given by a network of globally coupled excitatory neurons, where the coupling strength of each neuron to the mean field is non-uniform. Our key result is that stable irregular spiking states emerge even when only a minimal level of variability is present at a network level.

Another point we would like to stress here is that asynchronous firing states may be stabilized in the absence of external inputs. In the case here studied, there is an additional “difficulty” to the problem, in the sense that the pulse-coupled units considered are in excitable states, remaining quiet without sufficient drive from the other units in the network. The observed sustained asynchronous activity is hence self-organized.

We characterize how the features of the network dynamics depend on the coupling properties of the network and, in particular, we explore the possibility of chaos in the here studied case of excitable units, when partial synchrony is present, since this link has already been established in the case of coupled oscillators with a distribution of natural frequencies (Miritello et al.,

In the current work we study the properties of the self-induced stationary dynamical states in autonomous networks of excitable integrate-and-fire neurons. The neurons considered are characterized by a continuous state variable

where _{ex} = 0 mV represents the excitatory reversal potential and τ = 20 ms is the membrane time constant. Whenever the membrane potential reaches the threshold _{θ} = −50 mV, the discrete state of the neuron is set to _{i} = 1 for the duration of the spike. The voltage is reset, in addition, to its resting value of _{rest} = −60 mV, where it remains fixed for a refractory period of _{ref} = 5 ms. Equation (2) is not computed during the refractory period. Except for the times of spike occurrences, the discrete state of the neuron remains _{i} = 0 (no spike).

The conductance _{i} in Equation (2) integrates the influence of the time series of presynaptic spikes, decaying on the other side in absence of inputs:

where τ_{ex} = 5 ms is the conductance time constant. Incoming spikes from the _{i}_{i}_{i}

Here the synaptic weights _{ij}_{i}

Different connectivity structures are usually employed in the study of coupled oscillators, ranging from purely local rules to global couplings (Kuramoto and Battogtokh,

which corresponds to a uniform connectivity matrix without self coupling. All couplings are excitatory. The update rule (Equation 4) for the conductance upon presynaptic spiking then take the form:

where _{i}

We are interested in studying networks with non-uniform _{i}_{i}

for the _{i}_{i}

Several aspects of our model, in particular the asynchronous drifting state, can be investigated analytically as a consequence of the global coupling structure (Equation 5), as shown in Section 3.1. All further results are obtained from numerical simulations, for which, if not otherwise stated, a timestep of 0.01 ms has been used. We have also set the spike duration to one time-step, although these two parameters can be modified separately if desired, with our results not depending on the choice of the time-step, while the spike width does introduce minor quantitative changes to the results, as later discussed.

As a first approach we compute the response of a neuron with coupling constant _{i}_{i}

where we have denoted with

obtaining the time _{i}_{θ}, when starting from the resting potential _{rest}:

with:

We note, that the threshold potential _{θ} is only reached, if _{i}_{i}_{θ}. For the

The spiking frequency is _{r} = _{i}_{i} = _{ref}

of the neurons. Equations (10) and (13) describe the asynchronous drifting state in the thermodynamic limit

We studied our model, as defined by Equations (2) and (3), numerically for networks with typically

Three examples, for

_{rank} = _{i} uniformly distributed between

The plateau present in the case Δ_{i}

In Figure

I | : | inactive, |

I+D | : | partially inactive and drifting, |

D | : | fully drifting (asynchronous), |

D+S | : | mixed, containing both drifting and synchronized components, and |

S | : | fully synchronized. |

Examples of the rate distributions present in the individual phases are presented in Figure

The phase diagram is presented in Figure

The dashed lines in Figures

For a stable (non-trivial) attractor to arise in a network composed only of excitatory neurons, some limitation mechanism needs to be at play. Otherwise one observes a bifurcation phenomenon, similar to that of branching problems, in which only a critical network in the thermodynamic limit could be stable (Gros, _{ref}_{ref}

In order to study the phase transitions between states D and D+S and between D+S and S, we will resort in the following section to adiabatic paths in phase space crossing these lines.

Here we study the nature of the phase transitions between different dynamical states in Figure _{rank}

We observe that the emergence of synchronized clusters, the transition D → (D+S), is completely reversible. We believe this transition to be of second order and that the small discontinuity in the respective firing rate distributions observed in Figure

The disappearance of a subset of drifting neurons, the transition S → (D+S) is, on the other hand, not reversible. In this case, when

In networks of spiking neurons, it is essential to characterize not only the rate distribution of the system, but also the neurons' interspike-time statistics (Perkel et al., _{i}_{i}_{i}_{i}

D : The input received by a given neuron

D+S : The input received for drifting neuron _{i}

S : _{i}_{i}

As a frequently used measure of the regularity of a time distribution we have included in Figure

Of interest here are the finite

The high variability of the spiking intervals observed in the mixed state, as presented in Figure

Alternatively to a numerical evaluation of the Lyapunov exponents (a demanding task for large networks of spiking neurons), a somewhat more direct understanding of the chaotic state can be obtained by studying the relation between consecutive spike intervals. In Figure _{i}_{i}_{i}

_{i}_{i}

With a blue line we follow in Figure _{i}_{i}

We have determined the fractal dimension of the attracting set of pairs of spike intervals in the mixed phase by overlaying the attractor with a grid of 2^{r} × 2^{r} squares. For this calculation we employed a longer simulation with _{spikes}_{F}_{F}_{F}

_{box}^{r} × 2^{r} squares is laid upon the attracting set shown in Figure _{F}_{box}_{spikes}_{2}(_{box}_{2}(_{spikes}

We believe that the chaotic state arising in the mixed D+S state may be understood in analogy to the occurrence of chaos in the periodically driven pendulum (d'Humieres et al., _{syn}_{K}_{syn}_{syn}_{K}

In order to evaluate the robustness and the generality of the results here presented, we have evaluated the effects occurring when changing the size of the network and when allowing for variability in the connectivity matrix _{ij}

In Figure _{i}

In the previous sections, we considered the uniform connectivity matrix described by Equation (5). This allowed us to formulate the problem in terms of a mean-field coupling. We now analyze the robustness of the states found when a certain degree of variability is present in the weight matrix, viz when an extra variability term η is present:

Here we consider η to be drawn from a flat distribution with zero mean and a width Δ_{rank}

Finally, we test the robustness of the drifting state when perturbed with an external stimulus. To determine the stability of the state, we adiabatically increase the external stimulus _{ext}_{ext}_{ext}

_{ext}

In the present work we have studied a network of excitable units, consisting exclusively of excitatory neurons. In absence of external stimulus, the network is only able to remain active through its own activity, in a self-organized fashion. Below a certain average coupling strength of the network the activity dies out, whereas, if the average coupling is strong enough, the excitable units will collectively behave as pulse-coupled oscillators.

We have shown how the variability of coupling strengths determines the synchronization characteristics of the network, ranging from fully asynchronous, to fully synchronous activity. Interestingly, this variability, together with the neurons' refractoriness, is enough to keep the neural activity from exploding.

While we have initially assumed a purely mean field coupling (by setting all the synaptic weights _{ij}_{i}_{ij}s

Finally, we have studied the time structure of spikes in the different dynamical states observed. It is in the time domain that we find the main difference with natural neural networks. Spiking in real neurons is usually irregular, and it is often modeled as Poissonian, whereas in our network we found a very high degree of regularity, even in the asynchronous state. Only in the partially synchronous state we found a higher degree of variability, as a result from chaotic behavior. We have determined the fractal dimension of the respective strange attractor in the space of pairs of consecutive interspike intervals, finding fractional values of roughly 1.8 for the different neurons in the state.

While it has been often stated that inhibition is a necessary condition for bounded and uncorrelated activity, we have show here that uncorrelated aperiodic (and even chaotic) activity can be obtained with a network of excitatory-only connections, in a stable fashion and without external input. We are of course aware that the firing rates obtained in our simulations are high compared to

We have shown here that autonomous activity (sustained even in the absence of external inputs) may arise in networks of coupled excitable units, viz for units which are not intrinsically oscillating. We have also proposed a new tool to study the appearance of chaos in spiking neural networks by applying a box counting method to consecutive pairs of inter-spike intervals from a single unit. This tool is readily applicable both to experimental data and to the results of theory simulations in general.

RE carried out the simulations and produced the figures. CG guided the project and provided the theoretical framework. Both authors contributed to the writing of the article.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The support of the German Science Foundation (DFG) and the German Academic Exchange Service (DAAD) are acknowledged.