Dynamics Of Spiking Neural Networks Research Articles

Identifying steady state in the network dynamics of spiking neural networks

Analysis of the dynamics of complex networks can provide valuable information. For example, the dynamics can be used to characterize and differentiate between different network inputs and configurations. However, without quantitatively delineating the network's dynamic regimes, analysis of the network's dynamics is based on heuristics and qualitative signatures of transient or steady-state regimes. This is not ideal because interesting phenomena can occur during the transient regime, steady-state regime, or at the transition between the two dynamic regimes. Moreover, for simulated and observed systems, precise knowledge of the network's dynamical regime is imperative when considering metrics on minimal mathematical descriptions of the dynamics, otherwise either too much or too little data is analyzed. Here, we develop quantitative methods to ascertain the starting point and period of steady-state network activity. Using the precise knowledge of the network's dynamic regimes, we build minimal representations of the network dynamics that form the basis for future work. We show applications of our techniques on idealized signals and on the dynamics of a biologically inspired spiking neural network.

Identifying Steady State in the Network Dynamics of Spiking Neural Networks

Identifying Steady State in the Network Dynamics of Spiking Neural Networks

Mean-field approximations of networks of spiking neurons with short-term synaptic plasticity.

Low-dimensional descriptions of spiking neural network dynamics are an effective tool for bridging different scales of organization of brain structure and function. Recent advances in deriving mean-field descriptions for networks of coupled oscillators have sparked the development of a new generation of neural mass models. Of notable interest are mean-field descriptions of all-to-all coupled quadratic integrate-and-fire (QIF) neurons, which have already seen numerous extensions and applications. These extensions include different forms of short-term adaptation considered to play an important role in generating and sustaining dynamic regimes of interest in the brain. It is an open question, however, whether the incorporation of presynaptic forms of synaptic plasticity driven by single neuron activity would still permit the derivation of mean-field equations using the same method. Here we discuss this problem using an established model of short-term synaptic plasticity at the single neuron level, for which we present two different approaches for the derivation of the mean-field equations. We compare these models with a recently proposed mean-field approximation that assumes stochastic spike timings. In general, the latter fails to accurately reproduce the macroscopic activity in networks of deterministic QIF neurons with distributed parameters. We show that the mean-field models we propose provide a more accurate description of the network dynamics, although they are mathematically more involved. Using bifurcation analysis, we find that QIF networks with presynaptic short-term plasticity can express regimes of periodic bursting activity as well as bistable regimes. Together, we provide novel insight into the macroscopic effects of short-term synaptic plasticity in spiking neural networks, as well as two different mean-field descriptions for future investigations of such networks.

Collective and synchronous dynamics of photonic spiking neurons

Nonlinear dynamics of spiking neural networks have recently attracted much interest as an approach to understand possible information processing in the brain and apply it to artificial intelligence. Since information can be processed by collective spiking dynamics of neurons, the fine control of spiking dynamics is desirable for neuromorphic devices. Here we show that photonic spiking neurons implemented with paired nonlinear optical oscillators can be controlled to generate two modes of bio-realistic spiking dynamics by changing optical-pump amplitude. When the photonic neurons are coupled in a network, the interaction between them induces an effective change in the pump amplitude depending on the order parameter that characterizes synchronization. The experimental results show that the effective change causes spontaneous modification of the spiking modes and firing rates of clustered neurons, and such collective dynamics can be utilized to realize efficient heuristics for solving NP-hard combinatorial optimization problems.

Dynamical properties of spiking neural networks with small world topologies

Spiking neural networks can exhibit complex firing regimes whose characteristics are influenced by network topology. This paper is part of an investigation into the dynamical properties of spiking neural networks generated with small world topologies in comparison to those generated with Erdos-Renyi random graphs. Specifically, the parameters for small world and random graph network topology generation are tested empirically to find values which give rise to stable (fixed or periodic) vs. unstable or dissipative firing patterns. Similar to Erdos-Renyi random graph topologies, a critical threshold was found where the parameters of small world network generation lead to stable rather than dissipative patterns. Optimal parameters are identified for both small world and Erdos-Renyi random graph topologies which allow for stable firing patterns with minimal synapses. These results suggest questions that will form the basis for further research into the effects of topology class on firing dynamics of spiking neural networks.

Theta-Nested Gamma Oscillations in Next Generation Neural Mass Models.

Theta-nested gamma oscillations have been reported in many areas of the brain and are believed to represent a fundamental mechanism to transfer information across spatial and temporal scales. In a series of recent experiments in vitro it has been possible to replicate with an optogenetic theta frequency stimulation several features of cross-frequency coupling (CFC) among theta and gamma rhythms observed in behaving animals. In order to reproduce the main findings of these experiments we have considered a new class of neural mass models able to reproduce exactly the macroscopic dynamics of spiking neural networks. In this framework, we have examined two set-ups able to support collective gamma oscillations: namely, the pyramidal interneuronal network gamma (PING) and the interneuronal network gamma (ING). In both set-ups we observe the emergence of theta-nested gamma oscillations by driving the system with a sinusoidal theta-forcing in proximity of a Hopf bifurcation. These mixed rhythms always display phase amplitude coupling. However, two different types of nested oscillations can be identified: one characterized by a perfect phase locking between theta and gamma rhythms, corresponding to an overall periodic behavior; another one where the locking is imperfect and the dynamics is quasi-periodic or even chaotic. From our analysis it emerges that the locked states are more frequent in the ING set-up. In agreement with the experiments, we find theta-nested gamma oscillations for forcing frequencies in the range [1:10] Hz, whose amplitudes grow proportionally to the forcing intensity and which are clearly modulated by the theta phase. Furthermore, analogously to the experiments, the gamma power and the frequency of the gamma-power peak increase with the forcing amplitude. At variance with experimental findings, the gamma-power peak does not shift to higher frequencies by increasing the theta frequency. This effect can be obtained, in our model, only by incrementing, at the same time, also the stimulation power. An effect achieved by increasing the amplitude either of the noise or of the forcing term proportionally to the theta frequency. On the basis of our analysis both the PING and the ING mechanism give rise to theta-nested gamma oscillations with almost identical features.

Breakdown of order preservation in symmetric oscillator networks with pulse-coupling

Symmetric networks of coupled dynamical units exhibit invariant subspaces with two or more units synchronized. In time-continuously coupled systems, these invariant sets constitute barriers for the dynamics. For networks of units with local dynamics defined on the real line, this implies that the units' ordering is preserved and that their winding number is identical. Here, we show that in permutation-symmetric networks with pulse-coupling, the order is often no longer preserved. We analytically study a class of pulse-coupled oscillators (characterizing for instance the dynamics of spiking neural networks) and derive quantitative conditions for the breakdown of order preservation. We find that in general pulse-coupling yields additional dimensions to the state space such that units may change their order by avoiding the invariant sets. We identify a system of two symmetrically pulse-coupled identical oscillators where, contrary to intuition, the oscillators' average frequencies and thus their winding numbers are different.

Designing the Dynamics of Spiking Neural Networks

Precise timing of spikes and temporal locking are key elements of neural computation. Here we demonstrate how even strongly heterogeneous, deterministic neural networks with delayed interactions and complex topology can exhibit periodic patterns of spikes that are precisely timed. We develop an analytical method to find the set of all networks exhibiting a predefined pattern dynamics. Such patterns may be arbitrarily long and of complicated temporal structure. We point out that the same pattern can exist in very different networks and have different stability properties.