In a first part, we consider a single neuron which receives synaptic input
from a large number of other neurons. Neurophysiological recordings of the
membrane potential in the neuron look very much like trajectories of a
diffusion process. We model the membrane potential in the neuron by a
Cox-Ingersoll-Ross (CIR) type diffusion: assumed non-time homogeneous, this
diffusion can convey deterministic signals, in terms of time-varying
expectation. When the membrane potential in the neuron exceeds an excitation
threshold, the neuron generates spikes, thus transmitting information to other
neurons. The spike train generated by the neuron is modelled as a Poisson
point process with compensator proportional to the time the membrane potential
spends beyond the excitation threshold.
I a second part, we consider a large system of neurons $i=1,\dots,N$ processing the same deterministic signal. The main result is a functional central limit theorem for the collection of spike trains sent out by the neurons $i=1,\ldots,N$. In the covariance kernel of the limiting Gaussian process, a term appears which measures dependency in the (non-time homogeneous CIR) process of membrane potential. The pooled spike train allows to recover the original signal, up to some transformation which is explicit.
Some variants, including e.g deterministic refractory times, are also considered. For these, simulations indicate that the result above, valid for Poisson spike trains, remains a good guideline in a large neighborhood of the Poisson model. In cooperation with neurophysiologists at Mainz, we plan to check models of this type and to estimate model parameters through real data (intracellular measurements of the membrane potential, and recording of the spike trains sent out by the neuron).