Limits...
Dynamic Control of Synchronous Activity in Networks of Spiking Neurons

View Article: PubMed Central - PubMed

ABSTRACT

Oscillatory brain activity is believed to play a central role in neural coding. Accumulating evidence shows that features of these oscillations are highly dynamic: power, frequency and phase fluctuate alongside changes in behavior and task demands. The role and mechanism supporting this variability is however poorly understood. We here analyze a network of recurrently connected spiking neurons with time delay displaying stable synchronous dynamics. Using mean-field and stability analyses, we investigate the influence of dynamic inputs on the frequency of firing rate oscillations. We show that afferent noise, mimicking inputs to the neurons, causes smoothing of the system’s response function, displacing equilibria and altering the stability of oscillatory states. Our analysis further shows that these noise-induced changes cause a shift of the peak frequency of synchronous oscillations that scales with input intensity, leading the network towards critical states. We lastly discuss the extension of these principles to periodic stimulation, in which externally applied driving signals can trigger analogous phenomena. Our results reveal one possible mechanism involved in shaping oscillatory activity in the brain and associated control principles.

No MeSH data available.


Related in: MedlinePlus

Frequency transitions in a random network of spiking neurons.A. Schematic illustration depicting some features of the network model, in which interconnected cells are driven by independent sources of noise. Individual cells are connected via excitatory (red) and inhibitory (blue) synaptic connections. B. Synaptic connectivity matrix. Weights are randomly distributed around a mean value g (See Eq 2). C. Sample network activity, in which neurons spike timing is modulated by global, slow-wave synchronous oscillations in both low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Faster and more irregular firing modulations characterize the high-input state. D. Power spectral density of the network mean activity  in low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Other parameters are α = 100Hz, β = 300mV, g = −10mV/Hz, s = 20mV/Hz, τ = 25ms.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5036852&req=5

pone.0161488.g001: Frequency transitions in a random network of spiking neurons.A. Schematic illustration depicting some features of the network model, in which interconnected cells are driven by independent sources of noise. Individual cells are connected via excitatory (red) and inhibitory (blue) synaptic connections. B. Synaptic connectivity matrix. Weights are randomly distributed around a mean value g (See Eq 2). C. Sample network activity, in which neurons spike timing is modulated by global, slow-wave synchronous oscillations in both low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Faster and more irregular firing modulations characterize the high-input state. D. Power spectral density of the network mean activity in low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Other parameters are α = 100Hz, β = 300mV, g = −10mV/Hz, s = 20mV/Hz, τ = 25ms.

Mentions: In the present work, we analyze the dynamics of a generic network of spiking neurons (Fig 1A) whose membrane potential ui(t) evolves according to the following set of non-linear differential equationsα−1ddtui(t)=−ui(t)+N−1Σj=1NwijXj(t−τ)+Ii(t)(1)where α is the membrane time constant, wij = [W]ij are synaptic weights and where τ is a mean conduction delay. The membrane potential ui(t) represents the deviation from the neurons resting potential that is present in the absence of synaptic and external input. The presynaptic spike trains obey the non-homogenous Poisson processes Xi → Poisson(f[ui]) with rate f and the Dirac distribution δ(t). The firing rate function f has a non-linear sigmoid shape and is defined by f[ui] = (1 + exp[−βui])−1,i.e. the maximum firing rate approaches f = 1 for large membrane potentials. All neurons are subjected to afferent pre-synaptic inputs. The synaptic connectivity scheme was randomly set (Fig 1B), such thatwij=g+s ηij,(2)where g is the mean synaptic strength, s is the weight variance and nij are zero-mean independent Gaussian white noise such that < nijnkl >NxN = δikδjl with the Kronecker symbol δnm and where < >NxN is an average evaluated over all possible pairs of indices of the matrix W. We also consider other sources of synaptic inputs which we model as stochastic elements given the independent Gaussian white noise processes ξi with zero mean and < ξiξj >T = δij, where < >T is an average evaluated over an epoch of duration T or over an ensemble of realizations of processes. Such noise is meant to represent the effect of synaptic bombardment on neural membrane potentials. Such inputs can be shown to be well approximated by Gaussian processes in the diffusion limit case [23,24] and this is the approach we use in the following analysis. In the present work, we also consider a network of neurons whose spatial mean synaptic action is inhibitory with < 0.


Dynamic Control of Synchronous Activity in Networks of Spiking Neurons
Frequency transitions in a random network of spiking neurons.A. Schematic illustration depicting some features of the network model, in which interconnected cells are driven by independent sources of noise. Individual cells are connected via excitatory (red) and inhibitory (blue) synaptic connections. B. Synaptic connectivity matrix. Weights are randomly distributed around a mean value g (See Eq 2). C. Sample network activity, in which neurons spike timing is modulated by global, slow-wave synchronous oscillations in both low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Faster and more irregular firing modulations characterize the high-input state. D. Power spectral density of the network mean activity  in low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Other parameters are α = 100Hz, β = 300mV, g = −10mV/Hz, s = 20mV/Hz, τ = 25ms.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5036852&req=5

pone.0161488.g001: Frequency transitions in a random network of spiking neurons.A. Schematic illustration depicting some features of the network model, in which interconnected cells are driven by independent sources of noise. Individual cells are connected via excitatory (red) and inhibitory (blue) synaptic connections. B. Synaptic connectivity matrix. Weights are randomly distributed around a mean value g (See Eq 2). C. Sample network activity, in which neurons spike timing is modulated by global, slow-wave synchronous oscillations in both low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Faster and more irregular firing modulations characterize the high-input state. D. Power spectral density of the network mean activity in low (grey; D = 0.01) and high (blue; D = 0.50) input conditions. Other parameters are α = 100Hz, β = 300mV, g = −10mV/Hz, s = 20mV/Hz, τ = 25ms.
Mentions: In the present work, we analyze the dynamics of a generic network of spiking neurons (Fig 1A) whose membrane potential ui(t) evolves according to the following set of non-linear differential equationsα−1ddtui(t)=−ui(t)+N−1Σj=1NwijXj(t−τ)+Ii(t)(1)where α is the membrane time constant, wij = [W]ij are synaptic weights and where τ is a mean conduction delay. The membrane potential ui(t) represents the deviation from the neurons resting potential that is present in the absence of synaptic and external input. The presynaptic spike trains obey the non-homogenous Poisson processes Xi → Poisson(f[ui]) with rate f and the Dirac distribution δ(t). The firing rate function f has a non-linear sigmoid shape and is defined by f[ui] = (1 + exp[−βui])−1,i.e. the maximum firing rate approaches f = 1 for large membrane potentials. All neurons are subjected to afferent pre-synaptic inputs. The synaptic connectivity scheme was randomly set (Fig 1B), such thatwij=g+s ηij,(2)where g is the mean synaptic strength, s is the weight variance and nij are zero-mean independent Gaussian white noise such that < nijnkl >NxN = δikδjl with the Kronecker symbol δnm and where < >NxN is an average evaluated over all possible pairs of indices of the matrix W. We also consider other sources of synaptic inputs which we model as stochastic elements given the independent Gaussian white noise processes ξi with zero mean and < ξiξj >T = δij, where < >T is an average evaluated over an epoch of duration T or over an ensemble of realizations of processes. Such noise is meant to represent the effect of synaptic bombardment on neural membrane potentials. Such inputs can be shown to be well approximated by Gaussian processes in the diffusion limit case [23,24] and this is the approach we use in the following analysis. In the present work, we also consider a network of neurons whose spatial mean synaptic action is inhibitory with < 0.

View Article: PubMed Central - PubMed

ABSTRACT

Oscillatory brain activity is believed to play a central role in neural coding. Accumulating evidence shows that features of these oscillations are highly dynamic: power, frequency and phase fluctuate alongside changes in behavior and task demands. The role and mechanism supporting this variability is however poorly understood. We here analyze a network of recurrently connected spiking neurons with time delay displaying stable synchronous dynamics. Using mean-field and stability analyses, we investigate the influence of dynamic inputs on the frequency of firing rate oscillations. We show that afferent noise, mimicking inputs to the neurons, causes smoothing of the system&rsquo;s response function, displacing equilibria and altering the stability of oscillatory states. Our analysis further shows that these noise-induced changes cause a shift of the peak frequency of synchronous oscillations that scales with input intensity, leading the network towards critical states. We lastly discuss the extension of these principles to periodic stimulation, in which externally applied driving signals can trigger analogous phenomena. Our results reveal one possible mechanism involved in shaping oscillatory activity in the brain and associated control principles.

No MeSH data available.


Related in: MedlinePlus