Coding of time-dependent stimuli in homogeneous and heterogeneous neural populations.

We compare the information transmission of a time-dependent signal by two types of uncoupled neuron populations that differ in their sources of variability: i) a homogeneous population whose units receive independent noise and ii) a deterministic heterogeneous population, where each unit exhibits a different baseline firing rate ('disorder'). Our criterion for making both sources of variability quantitatively comparable is that the interspike-interval distributions are identical for both systems. Numerical simulations using leaky integrate-and-fire neurons unveil that a non-zero amount of both noise or disorder maximizes the encoding efficiency of the homogeneous and heterogeneous system, respectively, as a particular case of suprathreshold stochastic resonance. Our findings thus illustrate that heterogeneity can render similarly profitable effects for neuronal populations as dynamic noise. The optimal noise/disorder depends on the system size and the properties of the stimulus such as its intensity or cutoff frequency. We find that weak stimuli are better encoded by a noiseless heterogeneous population, whereas for strong stimuli a homogeneous population outperforms an equivalent heterogeneous system up to a moderate noise level. Furthermore, we derive analytical expressions of the coherence function for the cases of very strong noise and of vanishing intrinsic noise or heterogeneity, which predict the existence of an optimal noise intensity. Our results show that, depending on the type of signal, noise as well as heterogeneity can enhance the encoding performance of neuronal populations.

Synchronous spikes are necessary but not sufficient for a synchrony code in populations of spiking neurons.

Synchronous activity in populations of neurons potentially encodes special stimulus features. Selective readout of either synchronous or asynchronous activity allows formation of two streams of information processing. Theoretical work predicts that such a synchrony code is a fundamental feature of populations of spiking neurons if they operate in specific noise and stimulus regimes. Here we experimentally test the theoretical predictions by quantifying and comparing neuronal response properties in tuberous and ampullary electroreceptor afferents of the weakly electric fish Apteronotus leptorhynchus These related systems show similar levels of synchronous activity, but only in the more irregularly firing tuberous afferents a synchrony code is established, whereas in the more regularly firing ampullary afferents it is not. The mere existence of synchronous activity is thus not sufficient for a synchrony code. Single-cell features such as the irregularity of spiking and the frequency dependence of the neuron's transfer function determine whether synchronous spikes possess a distinct meaning for the encoding of time-dependent signals.

Partial synchronous output of a neuronal population under weak common noise: Analytical approaches to the correlation statistics.

We consider a homogeneous population of stochastic neurons that are driven by weak common noise (stimulus). To capture and analyze the joint firing events within the population, we introduce the partial synchronous output of the population. This is a time series defined by the events that at least a fixed fraction γ∈[0,1] of the population fires simultaneously within a small time interval. For this partial synchronous output we develop two analytical approaches to the correlation statistics. In the Gaussian approach we represent the synchronous output as a nonlinear transformation of the summed population activity and approximate the latter by a Gaussian process. In the combinatorial approach the synchronous output is represented by products of box-filtered spike trains of the single neurons. In both approaches we use linear-response theory to derive approximations for statistical measures that hold true for weak common noise. In particular, we calculate the mean value and power spectrum of the synchronous output and the cross-spectrum between synchronous output and common noise. We apply our results to the leaky integrate-and-fire neuron model and compare them to numerical simulations. The combinatorial approach is shown to provide a more accurate description of the statistics for small populations, whereas the Gaussian approximation yields compact formulas that work well for a sufficiently large population size. In particular, in the Gaussian approximation all statistical measures reveal a symmetry in the synchrony threshold γ around the mean value of the population activity. Our results may contribute to a better understanding of the role of coincidence detection in neural signal processing.

Spike-count distribution in a neuronal population under weak common stimulation.

We study the probability distribution of the number of synchronous action potentials (spike count) in a model network consisting of a homogeneous neural population that is driven by a common time-dependent stimulus. We derive two analytical approximations for the count statistics, which are based on linear response theory and hold true for weak input correlations. Comparison to numerical simulations of populations of integrate-and-fire neurons in different parameter regimes reveals that our theory correctly predicts how much a weak common stimulus increases the probability of common firing and of common silence in the neural population.

Biased diffusion inside regular islands under random symplectic perturbations.

We study the random concatenation of slightly different two-dimensional Hamiltonian maps with a mixed phase space. We consider a regular island whose fixed point is identical for all maps. Trajectories of the concatenated maps near this fixed point are no longer confined to invariant tori. We derive a stochastic model for the distance from the fixed point, which turns out to be a biased random walk with multiplicative noise. We give an analytical prediction of the survival probability of trajectories inside the regular island, which asymptotically is the product of a power law and an exponential. We confirm these results numerically for the parametrically perturbed standard map.