Escape rate models for noisy integrate-and-free neurons

Exploiting noise as a resource for computation and learning in spiking neural networks

Networks of spiking neurons underpin the extraordinary information-processing capabilities of the brain and have become pillar models in neuromorphic artificial intelligence. Despite extensive research on spiking neural networks (SNNs), most studies are established on deterministic models, overlooking the inherent non-deterministic, noisy nature of neural computations. This study introduces the noisy SNN (NSNN) and the noise-driven learning (NDL) rule by incorporating noisy neuronal dynamics to exploit the computational advantages of noisy neural processing. The NSNN provides a theoretical framework that yields scalable, flexible, and reliable computation and learning. We demonstrate that this framework leads to spiking neural models with competitive performance, improved robustness against challenging perturbations compared with deterministic SNNs, and better reproducing probabilistic computation in neural coding. Generally, this study offers a powerful and easy-to-use tool for machine learning, neuromorphic intelligence practitioners, and computational neuroscience researchers.

Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons

The macroscopic dynamics of large populations of neurons can be mathematically analyzed using low-dimensional firing-rate or neural-mass models. However, these models fail to capture spike synchronization effects and nonstationary responses of the population activity to rapidly changing stimuli. Here we derive low-dimensional firing-rate models for homogeneous populations of neurons modeled as time-dependent renewal processes. The class of renewal neurons includes integrate-and-fire models driven by white noise and has been frequently used to model neuronal refractoriness and spike synchronization dynamics. The derivation is based on an eigenmode expansion of the associated refractory density equation, which generalizes previous spectral methods for Fokker-Planck equations to arbitrary renewal models. We find a simple relation between the eigenvalues characterizing the timescales of the firing rate dynamics and the Laplace transform of the interspike interval density, for which explicit expressions are available for many renewal models. Retaining only the first eigenmode already yields a reliable low-dimensional approximation of the firing-rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset. We explicitly demonstrate the validity of our model for a large homogeneous population of Poisson neurons with absolute refractoriness and other renewal models that admit an explicit analytical calculation of the eigenvalues. The eigenmode expansion presented here provides a systematic framework for alternative firing-rate models in computational neuroscience based on spiking neuron dynamics with refractoriness.

Improved similarity measures for small sets of spike trains

Multiple measures have been developed to quantify the similarity between two spike trains. These measures have been used for the quantification of the mismatch between neuron models and experiments as well as for the classification of neuronal responses in neuroprosthetic devices and electrophysiological experiments. Frequently only a few spike trains are available in each class. We derive analytical expressions for the small-sample bias present when comparing estimators of the time-dependent firing intensity. We then exploit analogies between the comparison of firing intensities and previously used spike train metrics and show that improved spike train measures can be successfully used for fitting neuron models to experimental data, for comparisons of spike trains, and classification of spike train data. In classification tasks, the improved similarity measures can increase the recovered information. We demonstrate that when similarity measures are used for fitting mathematical models, all previous methods systematically underestimate the noise. Finally, we show a striking implication of this deterministic bias by reevaluating the results of the single-neuron prediction challenge.

Mean, Variance, and Autocorrelation of Subthreshold Potential Fluctuations Driven by Filtered Conductance Shot Noise

We study the subthreshold voltage fluctuations of a conductance-based passive point neuron stimulated by filtered Poissonian shot noise. We give exact analytical expressions in terms of quadratures for the first two time-dependent moments and the autocorrelation function of the membrane voltage. We also derive simplified expressions for the moments in terms of elementary functions that hold true in the limit case of short filter time, small spike amplitude, and a single synaptic reversal potential. By means of these expressions, we show that for an ensemble of equilibrated conductances but sharp initial voltage (corresponding to a short voltage clamp at the initial time), the mean and the standard deviation can display nonmonotonic time courses. In particular, transient changes in the standard deviation disagree strongly with the predictions of the commonly used effective time constant approximation over a large parameter range. We also study the dependence of the correlation time of the voltage on the synaptic spike amplitude and the synaptic input rate. All results are confirmed by extensive stochastic simulations.

Firing time statistics for driven neuron models: analytic expressions versus numerics

Analytical expressions are put forward to investigate the forced spiking activity of abstract neuron models such as the driven leaky integrate-and-fire model. The method is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. The scheme yields excellent agreement with numerical Langevin and Fokker-Planck simulations of the full nonstationary dynamics, not only for the first-passage time statistics, but also for the important interspike interval (residence time) distribution.