3,854 research outputs found
Stochastic dynamics of a finite-size spiking neural network
We present a simple Markov model of spiking neural dynamics that can be
analytically solved to characterize the stochastic dynamics of a finite-size
spiking neural network. We give closed-form estimates for the equilibrium
distribution, mean rate, variance and autocorrelation function of the network
activity. The model is applicable to any network where the probability of
firing of a neuron in the network only depends on the number of neurons that
fired in a previous temporal epoch. Networks with statistically homogeneous
connectivity and membrane and synaptic time constants that are not excessively
long could satisfy these conditions. Our model completely accounts for the size
of the network and correlations in the firing activity. It also allows us to
examine how the network dynamics can deviate from mean-field theory. We show
that the model and solutions are applicable to spiking neural networks in
biophysically plausible parameter regimes
Metastable states and quasicycles in a stochastic Wilson-Cowan\ud model of neuronal population dynamics
We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N -> infinity, where N determines the size of each population, the dynamics is described by deterministic WilsonâCowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steadyâstate probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N, we calculate the exponentially small rate of noiseâinduced transitions between the resulting metastable states using a WentzelâKramersâBrillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory/inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles)
Mesoscopic description of hippocampal replay and metastability in spiking neural networks with short-term plasticity
Bottom-up models of functionally relevant patterns of neural activity provide
an explicit link between neuronal dynamics and computation. A prime example of
functional activity pattern is hippocampal replay, which is critical for memory
consolidation. The switchings between replay events and a low-activity state in
neural recordings suggests metastable neural circuit dynamics. As metastability
has been attributed to noise and/or slow fatigue mechanisms, we propose a
concise mesoscopic model which accounts for both. Crucially, our model is
bottom-up: it is analytically derived from the dynamics of finite-size networks
of Linear-Nonlinear Poisson neurons with short-term synaptic depression. As
such, noise is explicitly linked to spike noise and network size, and fatigue
is explicitly linked to synaptic dynamics. To derive the mesosocpic model, we
first consider a homogeneous spiking neural network and follow the temporal
coarse-graining approach of Gillespie ("chemical Langevin equation"), which can
be naturally interpreted as a stochastic neural mass model. The Langevin
equation is computationally inexpensive to simulate and enables a thorough
study of metastable dynamics in classical setups (population spikes and Up-Down
states dynamics) by means of phase-plane analysis. This stochastic neural mass
model is the basic component of our mesoscopic model for replay. We show that
our model faithfully captures the stochastic nature of individual replayed
trajectories. Moreover, compared to the deterministic Romani-Tsodyks model of
place cell dynamics, it exhibits a higher level of variability in terms of
content, direction and timing of replay events, compatible with biological
evidence and could be functionally desirable. This variability is the product
of a new dynamical regime where metastability emerges from a complex interplay
between finite-size fluctuations and local fatigue.Comment: 43 pages, 8 figure
Deterministic networks for probabilistic computing
Neural-network models of high-level brain functions such as memory recall and
reasoning often rely on the presence of stochasticity. The majority of these
models assumes that each neuron in the functional network is equipped with its
own private source of randomness, often in the form of uncorrelated external
noise. However, both in vivo and in silico, the number of noise sources is
limited due to space and bandwidth constraints. Hence, neurons in large
networks usually need to share noise sources. Here, we show that the resulting
shared-noise correlations can significantly impair the performance of
stochastic network models. We demonstrate that this problem can be overcome by
using deterministic recurrent neural networks as sources of uncorrelated noise,
exploiting the decorrelating effect of inhibitory feedback. Consequently, even
a single recurrent network of a few hundred neurons can serve as a natural
noise source for large ensembles of functional networks, each comprising
thousands of units. We successfully apply the proposed framework to a diverse
set of binary-unit networks with different dimensionalities and entropies, as
well as to a network reproducing handwritten digits with distinct predefined
frequencies. Finally, we show that the same design transfers to functional
networks of spiking neurons.Comment: 22 pages, 11 figure
Intrinsically-generated fluctuating activity in excitatory-inhibitory networks
Recurrent networks of non-linear units display a variety of dynamical regimes
depending on the structure of their synaptic connectivity. A particularly
remarkable phenomenon is the appearance of strongly fluctuating, chaotic
activity in networks of deterministic, but randomly connected rate units. How
this type of intrinsi- cally generated fluctuations appears in more realistic
networks of spiking neurons has been a long standing question. To ease the
comparison between rate and spiking networks, recent works investigated the
dynami- cal regimes of randomly-connected rate networks with segregated
excitatory and inhibitory populations, and firing rates constrained to be
positive. These works derived general dynamical mean field (DMF) equations
describing the fluctuating dynamics, but solved these equations only in the
case of purely inhibitory networks. Using a simplified excitatory-inhibitory
architecture in which DMF equations are more easily tractable, here we show
that the presence of excitation qualitatively modifies the fluctuating activity
compared to purely inhibitory networks. In presence of excitation,
intrinsically generated fluctuations induce a strong increase in mean firing
rates, a phenomenon that is much weaker in purely inhibitory networks.
Excitation moreover induces two different fluctuating regimes: for moderate
overall coupling, recurrent inhibition is sufficient to stabilize fluctuations,
for strong coupling, firing rates are stabilized solely by the upper bound
imposed on activity, even if inhibition is stronger than excitation. These
results extend to more general network architectures, and to rate networks
receiving noisy inputs mimicking spiking activity. Finally, we show that
signatures of the second dynamical regime appear in networks of
integrate-and-fire neurons
Metastability in a stochastic neural network modeled as a velocity jump Markov process
One of the major challenges in neuroscience is to determine how noise that is
present at the molecular and cellular levels affects dynamics and information
processing at the macroscopic level of synaptically coupled neuronal
populations. Often noise is incorprated into deterministic network models using
extrinsic noise sources. An alternative approach is to assume that noise arises
intrinsically as a collective population effect, which has led to a master
equation formulation of stochastic neural networks. In this paper we extend the
master equation formulation by introducing a stochastic model of neural
population dynamics in the form of a velocity jump Markov process. The latter
has the advantage of keeping track of synaptic processing as well as spiking
activity, and reduces to the neural master equation in a particular limit. The
population synaptic variables evolve according to piecewise deterministic
dynamics, which depends on population spiking activity. The latter is
characterised by a set of discrete stochastic variables evolving according to a
jump Markov process, with transition rates that depend on the synaptic
variables. We consider the particular problem of rare transitions between
metastable states of a network operating in a bistable regime in the
deterministic limit. Assuming that the synaptic dynamics is much slower than
the transitions between discrete spiking states, we use a WKB approximation and
singular perturbation theory to determine the mean first passage time to cross
the separatrix between the two metastable states. Such an analysis can also be
applied to other velocity jump Markov processes, including stochastic
voltage-gated ion channels and stochastic gene networks
Integration of continuous-time dynamics in a spiking neural network simulator
Contemporary modeling approaches to the dynamics of neural networks consider
two main classes of models: biologically grounded spiking neurons and
functionally inspired rate-based units. The unified simulation framework
presented here supports the combination of the two for multi-scale modeling
approaches, the quantitative validation of mean-field approaches by spiking
network simulations, and an increase in reliability by usage of the same
simulation code and the same network model specifications for both model
classes. While most efficient spiking simulations rely on the communication of
discrete events, rate models require time-continuous interactions between
neurons. Exploiting the conceptual similarity to the inclusion of gap junctions
in spiking network simulations, we arrive at a reference implementation of
instantaneous and delayed interactions between rate-based models in a spiking
network simulator. The separation of rate dynamics from the general connection
and communication infrastructure ensures flexibility of the framework. We
further demonstrate the broad applicability of the framework by considering
various examples from the literature ranging from random networks to neural
field models. The study provides the prerequisite for interactions between
rate-based and spiking models in a joint simulation
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