58 research outputs found
Taming neuronal noise with large networks
How does reliable computation emerge from networks of noisy neurons? While individual neurons are intrinsically noisy, the collective dynamics of populations of neurons taken as a whole can be almost deterministic, supporting the hypothesis that, in the brain, computation takes place at the level of neuronal populations. Mathematical models of networks of noisy spiking neurons allow us to study the effects of neuronal noise on the dynamics of large networks. Classical mean-field models, i.e., models where all neurons are identical and where each neuron receives the average spike activity of the other neurons, offer toy examples where neuronal noise is absorbed in large networks, that is, large networks behave like deterministic systems. In particular, the dynamics of these large networks can be described by deterministic neuronal population equations. In this thesis, I first generalize classical mean-field limit proofs to a broad class of spiking neuron models that can exhibit spike-frequency adaptation and short-term synaptic plasticity, in addition to refractoriness. The mean-field limit can be exactly described by a multidimensional partial differential equation; the long time behavior of which can be rigorously studied using deterministic methods. Then, we show that there is a conceptual link between mean-field models for networks of spiking neurons and latent variable models used for the analysis of multi-neuronal recordings. More specifically, we use a recently proposed finite-size neuronal population equation, which we first mathematically clarify, to design a tractable Expectation-Maximization-type algorithm capable of inferring the latent population activities of multi-population spiking neural networks from the spike activity of a few visible neurons only, illustrating the idea that latent variable models can be seen as partially observed mean-field models. In classical mean-field models, neurons in large networks behave like independent, identically distributed processes driven by the average population activity -- a deterministic quantity, by the law of large numbers. The fact the neurons are identically distributed processes implies a form of redundancy that has not been observed in the cortex and which seems biologically implausible. To show, numerically, that the redundancy present in classical mean-field models is unnecessary for neuronal noise absorption in large networks, I construct a disordered network model where networks of spiking neurons behave like deterministic rate networks, despite the absence of redundancy. This last result suggests that the concentration of measure phenomenon, which generalizes the ``law of large numbers'' of classical mean-field models, might be an instrumental principle for understanding the emergence of noise-robust population dynamics in large networks of noisy neurons
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
Linear response for spiking neuronal networks with unbounded memory
We establish a general linear response relation for spiking neuronal
networks, based on chains with unbounded memory. This relation allows us to
predict the influence of a weak amplitude time-dependent external stimuli on
spatio-temporal spike correlations, from the spontaneous statistics (without
stimulus) in a general context where the memory in spike dynamics can extend
arbitrarily far in the past. Using this approach, we show how linear response
is explicitly related to neuronal dynamics with an example, the gIF model,
introduced by M. Rudolph and A. Destexhe. This example illustrates the
collective effect of the stimuli, intrinsic neuronal dynamics, and network
connectivity on spike statistics. We illustrate our results with numerical
simulations.Comment: 60 pages, 8 figure
From Spiking Neuron Models to Linear-Nonlinear Models
Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates
Autoregressive Point-Processes as Latent State-Space Models: a Moment-Closure Approach to Fluctuations and Autocorrelations
Modeling and interpreting spike train data is a task of central importance in
computational neuroscience, with significant translational implications. Two
popular classes of data-driven models for this task are autoregressive Point
Process Generalized Linear models (PPGLM) and latent State-Space models (SSM)
with point-process observations. In this letter, we derive a mathematical
connection between these two classes of models. By introducing an auxiliary
history process, we represent exactly a PPGLM in terms of a latent, infinite
dimensional dynamical system, which can then be mapped onto an SSM by basis
function projections and moment closure. This representation provides a new
perspective on widely used methods for modeling spike data, and also suggests
novel algorithmic approaches to fitting such models. We illustrate our results
on a phasic bursting neuron model, showing that our proposed approach provides
an accurate and efficient way to capture neural dynamics
Mean-field limit of age and leaky memory dependent Hawkes processes
We propose a mean-field model of interacting point processes where each process has a memory of the time elapsed since its last event (age) and its recent past (leaky memory), generalizing Age-dependent Hawkes processes. The model is motivated by interacting nonlinear Hawkes processes with Markovian self-interaction and networks of spiking neurons with adaptation and short-term synaptic plasticity.
By proving propagation of chaos and using a path integral representation for the law of the limit process, we show that, in the mean-field limit, the empirical measure of the system follows a multidimensional nonlocal transport equation
Fundamental activity constraints lead to specific interpretations of the connectome
The continuous integration of experimental data into coherent models of the
brain is an increasing challenge of modern neuroscience. Such models provide a
bridge between structure and activity, and identify the mechanisms giving rise
to experimental observations. Nevertheless, structurally realistic network
models of spiking neurons are necessarily underconstrained even if experimental
data on brain connectivity are incorporated to the best of our knowledge.
Guided by physiological observations, any model must therefore explore the
parameter ranges within the uncertainty of the data. Based on simulation
results alone, however, the mechanisms underlying stable and physiologically
realistic activity often remain obscure. We here employ a mean-field reduction
of the dynamics, which allows us to include activity constraints into the
process of model construction. We shape the phase space of a multi-scale
network model of the vision-related areas of macaque cortex by systematically
refining its connectivity. Fundamental constraints on the activity, i.e.,
prohibiting quiescence and requiring global stability, prove sufficient to
obtain realistic layer- and area-specific activity. Only small adaptations of
the structure are required, showing that the network operates close to an
instability. The procedure identifies components of the network critical to its
collective dynamics and creates hypotheses for structural data and future
experiments. The method can be applied to networks involving any neuron model
with a known gain function.Comment: J. Schuecker and M. Schmidt contributed equally to this wor
Dynamics and spike trains statistics in conductance-based Integrate-and-Fire neural networks with chemical and electric synapses
We investigate the effect of electric synapses (gap junctions) on collective
neuronal dynamics and spike statistics in a conductance-based
Integrate-and-Fire neural network, driven by a Brownian noise, where
conductances depend upon spike history. We compute explicitly the time
evolution operator and show that, given the spike-history of the network and
the membrane potentials at a given time, the further dynamical evolution can be
written in a closed form. We show that spike train statistics is described by a
Gibbs distribution whose potential can be approximated with an explicit
formula, when the noise is weak. This potential form encompasses existing
models for spike trains statistics analysis such as maximum entropy models or
Generalized Linear Models (GLM). We also discuss the different types of
correlations: those induced by a shared stimulus and those induced by neurons
interactions.Comment: 42 pages, 1 figure, submitte
Impact of network structure and cellular response on spike time correlations
Novel experimental techniques reveal the simultaneous activity of larger and
larger numbers of neurons. As a result there is increasing interest in the
structure of cooperative -- or correlated -- activity in neural populations,
and in the possible impact of such correlations on the neural code. A
fundamental theoretical challenge is to understand how the architecture of
network connectivity along with the dynamical properties of single cells shape
the magnitude and timescale of correlations. We provide a general approach to
this problem by extending prior techniques based on linear response theory. We
consider networks of general integrate-and-fire cells with arbitrary
architecture, and provide explicit expressions for the approximate
cross-correlation between constituent cells. These correlations depend strongly
on the operating point (input mean and variance) of the neurons, even when
connectivity is fixed. Moreover, the approximations admit an expansion in
powers of the matrices that describe the network architecture. This expansion
can be readily interpreted in terms of paths between different cells. We apply
our results to large excitatory-inhibitory networks, and demonstrate first how
precise balance --- or lack thereof --- between the strengths and timescales of
excitatory and inhibitory synapses is reflected in the overall correlation
structure of the network. We then derive explicit expressions for the average
correlation structure in randomly connected networks. These expressions help to
identify the important factors that shape coordinated neural activity in such
networks
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