Age Dependent Hawkes Process

In the last decade, Hawkes processes have received a lot of attention as good models for functional connectivity in neural spiking networks. In this paper we consider a variant of this process, the Age Dependent Hawkes process, which incorporates individual post-jump behaviour into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean field type, we study large network limits and establish the propagation of chaos property of the system.

Hypoelliptic diffusions: discretization, filtering and inference from complete and partial observations

The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, due to the noise structure, where the noise components of the different coordinates of the multi-dimensional process operate on different time scales, standard inference tools are ill conditioned. In this paper, we propose to use a higher order scheme to approximate the likelihood, such that the different time scales are appropriately accounted for. We show consistency and asymptotic normality with non-typical convergence rates. When only partial observations are available, we embed the approximation into a filtering algorithm for the unobserved coordinates, and use this as a building block in a Stochastic Approximation Expectation Maximization algorithm. We illustrate on simulated data from three models; the Harmonic Oscillator, the FitzHugh-Nagumo model used to model the membrane potential evolution in neuroscience, and the Synaptic Inhibition and Excitation model used for determination of neuronal synaptic input.

Multivariate inhomogeneous diffusion models with covariates and mixed effects

Modeling of longitudinal data often requires diffusion models that incorporate overall time-dependent, nonlinear dynamics of multiple components and provide sufficient flexibility for subject-specific modeling. This complexity challenges parameter inference and approximations are inevitable. We propose a method for approximate maximum-likelihood parameter estimation in multivariate time-inhomogeneous diffusions, where subject-specific flexibility is accounted for by incorporation of multidimensional mixed effects and covariates. We consider $N$ multidimensional independent diffusions $X^i = (X^i_t)_{0leq tleq T^i}, 1leq ileq N$, with common overall model structure and unknown fixed-effects parameter $mu$. Their dynamics differ by the subject-specific random effect $phi^i$ in the drift and possibly by (known) covariate information, different initial conditions and observation times and duration. The distribution of $phi^i$ is parametrized by an unknown $vartheta$ and $theta = (mu, vartheta)$ is the target of statistical inference. Its maximum likelihood estimator is derived from the continuous-time likelihood. We prove consistency and asymptotic normality of $hat{theta}_N$ when the number $N$ of subjects goes to infinity using standard techniques and consider the more general concept of local asymptotic normality for less regular models. The bias induced by time-discretization of sufficient statistics is investigated. We discuss verification of conditions and investigate parameter estimation and hypothesis testing in simulations.

The space-clamped Hodgkin-Huxley system with random synaptic input: inhibition of spiking by weak noise and analysis with moment equations

We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model system obtained by an Euler method, it is found that with excitation only there is a critical value of the steady state excitatory conductance for repetitive spiking without noise and for values of the conductance near the critical value small noise has a powerfully inhibitory effect. For a given level of inhibition there is also a critical value of the steady state excitatory conductance for repetitive firing and it is demonstrated that noise either in the excitatory or inhibitory processes or both can powerfully inhibit spiking. Furthermore, near the critical value, inverse stochastic resonance was observed when noise was present only in the inhibitory input process. The system of 27 coupled deterministic differential equations for the approximate first and second order moments of the 6-dimensional model is derived. The moment differential equations are solved using Runge-Kutta methods and the solutions are compared with the results obtained by simulation for various sets of parameters including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex. The mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels. In the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells the assunptions underlying the moment equation approach are not satisfied.

Multi-class oscillating systems of interacting neurons

We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems. We study situations in which the limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov processes and their diffusion approximations.

Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods

Parameter estimation in multidimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem. In neuroscience, the membrane potential evolution in single neurons can be measured at high frequency, but biophysical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris-Lecar model, defined by a nonlinear two-dimensional stochastic differential equation. The coordinates are coupled, that is, the unobserved coordinate is nonautonomous, the model exhibits oscillations to mimic the spiking behavior, which means it is not of gradient-type, and the measurement noise from intracellular recordings is typically negligible. Therefore, the hidden Markov model framework is degenerate, and available methods break down. The main contributions of this paper are an approach to estimate in this ill-posed situation and nonasymptotic convergence results for the method. Specifically, we propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parameters maximizing a pseudo-likelihood through a stochastic version of the Expectation-Maximization algorithm. It turns out that even the rate scaling parameter governing the opening and closing of ion channels of the unobserved coordinate can be reasonably estimated. An experimental data set of intracellular recordings of the membrane potential of a spinal motoneuron of a red-eared turtle is analyzed, and the performance is further evaluated in a simulation study.

The Morris-Lecar neuron model embeds a leaky integrate-and-fire model

We show that the stochastic Morris-Lecar neuron, in a neighborhood of its stable point, can be approximated by a two-dimensional Ornstein-Uhlenbeck (OU) modulation of a constant circular motion. The associated radial OU process is an example of a leaky integrate-and-fire (LIF) model prior to firing. A new model constructed from a radial OU process together with a simple firing mechanism based on detailed Morris-Lecar firing statistics reproduces the Morris-Lecar Interspike Interval (ISI) distribution, and has the computational advantages of a LIF. The result justifies the large amount of attention paid to the LIF models.

Practical Estimation of High Dimensional Stochastic Differential Mixed-Effects Models

Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrinsic randomness of a system around its drift can be identified and separated from the drift itself. When it is of interest to model dynamics within a given population, i.e. to model simultaneously the performance of several experiments or subjects, mixed-effects modelling allows for the distinction of between and within experiment variability. A framework to model dynamics within a population using SDEs is proposed, representing simultaneously several sources of variation: variability between experiments using a mixed-effects approach and stochasticity in the individual dynamics using SDEs. These stochastic differential mixed-effects models have applications in e.g. pharmacokinetics/pharmacodynamics and biomedical modelling. A parameter estimation method is proposed and computational guidelines for an efficient implementation are given. Finally the method is evaluated using simulations from standard models like the two-dimensional Ornstein-Uhlenbeck (OU) and the square root models.