90 research outputs found
The perfect integrator driven by Poisson input and its approximation in the diffusion limit
In this note we consider the perfect integrator driven by Poisson process
input. We derive its equilibrium and response properties and contrast them to
the approximations obtained by applying the diffusion approximation. In
particular, the probability density in the vicinity of the threshold differs,
which leads to altered response properties of the system in equilibrium.Comment: 7 pages, 3 figures, v2: corrected authors in referenc
Fluctuations and information filtering in coupled populations of spiking neurons with adaptation
Finite-sized populations of spiking elements are fundamental to brain
function, but also used in many areas of physics. Here we present a theory of
the dynamics of finite-sized populations of spiking units, based on a
quasi-renewal description of neurons with adaptation. We derive an integral
equation with colored noise that governs the stochastic dynamics of the
population activity in response to time-dependent stimulation and calculate the
spectral density in the asynchronous state. We show that systems of coupled
populations with adaptation can generate a frequency band in which sensory
information is preferentially encoded. The theory is applicable to fully as
well as randomly connected networks, and to leaky integrate-and-fire as well as
to generalized spiking neurons with adaptation on multiple time scales
Equilibrium and Response Properties of the Integrate-and-Fire Neuron in Discrete Time
The integrate-and-fire neuron with exponential postsynaptic potentials is a frequently employed model to study neural networks. Simulations in discrete time still have highest performance at moderate numerical errors, which makes them first choice for long-term simulations of plastic networks. Here we extend the population density approach to investigate how the equilibrium and response properties of the leaky integrate-and-fire neuron are affected by time discretization. We present a novel analytical treatment of the boundary condition at threshold, taking both discretization of time and finite synaptic weights into account. We uncover an increased membrane potential density just below threshold as the decisive property that explains the deviations found between simulations and the classical diffusion approximation. Temporal discretization and finite synaptic weights both contribute to this effect. Our treatment improves the standard formula to calculate the neuron's equilibrium firing rate. Direct solution of the Markov process describing the evolution of the membrane potential density confirms our analysis and yields a method to calculate the firing rate exactly. Knowing the shape of the membrane potential distribution near threshold enables us to devise the transient response properties of the neuron model to synaptic input. We find a pronounced non-linear fast response component that has not been described by the prevailing continuous time theory for Gaussian white noise input
Reconstruction of recurrent synaptic connectivity of thousands of neurons from simulated spiking activity
Dynamics and function of neuronal networks are determined by their synaptic
connectivity. Current experimental methods to analyze synaptic network
structure on the cellular level, however, cover only small fractions of
functional neuronal circuits, typically without a simultaneous record of
neuronal spiking activity. Here we present a method for the reconstruction of
large recurrent neuronal networks from thousands of parallel spike train
recordings. We employ maximum likelihood estimation of a generalized linear
model of the spiking activity in continuous time. For this model the point
process likelihood is concave, such that a global optimum of the parameters can
be obtained by gradient ascent. Previous methods, including those of the same
class, did not allow recurrent networks of that order of magnitude to be
reconstructed due to prohibitive computational cost and numerical
instabilities. We describe a minimal model that is optimized for large networks
and an efficient scheme for its parallelized numerical optimization on generic
computing clusters. For a simulated balanced random network of 1000 neurons,
synaptic connectivity is recovered with a misclassification error rate of less
than 1% under ideal conditions. We show that the error rate remains low in a
series of example cases under progressively less ideal conditions. Finally, we
successfully reconstruct the connectivity of a hidden synfire chain that is
embedded in a random network, which requires clustering of the network
connectivity to reveal the synfire groups. Our results demonstrate how synaptic
connectivity could potentially be inferred from large-scale parallel spike
train recordings.Comment: This is the final version of the manuscript from the publisher which
supersedes our original pre-print version. The spike data used in this paper
and the code that implements our connectivity reconstruction method are
publicly available for download at http://dx.doi.org/10.5281/zenodo.17662 and
http://dx.doi.org/10.5281/zenodo.17663 respectivel
Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size
Neural population equations such as neural mass or field models are widely used to study brain activity on a large scale. However, the relation of these models to the properties of single neurons is unclear. Here we derive an equation for several interacting populations at the mesoscopic scale starting from a microscopic model of randomly connected generalized integrate-and-fire neuron models. Each population consists of 50 -- 2000 neurons of the same type but different populations account for different neuron types. The stochastic population equations that we find reveal how spike-history effects in single-neuron dynamics such as refractoriness and adaptation interact with finite-size fluctuations on the population level. Efficient integration of the stochastic mesoscopic equations reproduces the statistical behavior of the population activities obtained from microscopic simulations of a full spiking neural network model. The theory describes nonlinear emergent dynamics like finite-size-induced stochastic transitions in multistable networks and synchronization in balanced networks of excitatory and inhibitory neurons. The mesoscopic equations are employed to rapidly simulate a model of a local cortical microcircuit consisting of eight neuron types. Our theory establishes a general framework for modeling finite-size neural population dynamics based on single cell and synapse parameters and offers an efficient approach to analyzing cortical circuits and computations
Network dynamics of spiking neurons with adaptation
How to describe circuit level cortical dynamics
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