Fredholm theory for the mean first-passage time of integrate-and-fire oscillators with colored noise input.

We develop a method to investigate the effect of noise timescales on the first-passage time of nonlinear oscillators. Using Fredholm theory, we derive an exact integral equation for the mean event rate of a leaky integrate-and-fire oscillator that receives constant input and temporally correlated noise. Furthermore, we show that Fredholm theory provides a unified framework to determine the system scaling behavior for small and large noise timescales. In this framework, the leading-order and higher-order asymptotic corrections for slow and fast noise are naturally emerging. We show the scaling behavior in the both limits is not reciprocal. We discuss further how this approach can be extended to study the first-passage time in a general class of nonlinear oscillators driven by colored noise at arbitrary timescales.

Complete Mean-Field Theory for Dynamics of Binary Recurrent Networks.

We develop a unified theory that encompasses the macroscopic dynamics of recurrent interactions of binary units within arbitrary network architectures. Using the martingale theory, our mathematical analysis provides a complete description of nonequilibrium fluctuations in networks with a finite size and finite degree of interactions. Our approach allows the investigation of systems for which a deterministic mean-field theory breaks down. To demonstrate this, we uncover a novel dynamic state in which a recurrent network of binary units with statistically inhomogeneous interactions, along with an asynchronous behavior, also exhibits collective nontrivial stochastic fluctuations in the thermodynamical limit.

A reanalysis of "Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons".

Neuronal activity in the central nervous system varies strongly in time and across neuronal populations. It is a longstanding proposal that such fluctuations generically arise from chaotic network dynamics. Various theoretical studies predict that the rich dynamics of rate models operating in the chaotic regime can subserve circuit computation and learning. Neurons in the brain, however, communicate via spikes and it is a theoretical challenge to obtain similar rate fluctuations in networks of spiking neuron models. A recent study investigated spiking balanced networks of leaky integrate and fire (LIF) neurons and compared their dynamics to a matched rate network with identical topology, where single unit input-output functions were chosen from isolated LIF neurons receiving Gaussian white noise input. A mathematical analogy between the chaotic instability in networks of rate units and the spiking network dynamics was proposed. Here we revisit the behavior of the spiking LIF networks and these matched rate networks. We find expected hallmarks of a chaotic instability in the rate network: For supercritical coupling strength near the transition point, the autocorrelation time diverges. For subcritical coupling strengths, we observe critical slowing down in response to small external perturbations. In the spiking network, we found in contrast that the timescale of the autocorrelations is insensitive to the coupling strength and that rate deviations resulting from small input perturbations rapidly decay. The decay speed even accelerates for increasing coupling strength. In conclusion, our reanalysis demonstrates fundamental differences between the behavior of pulse-coupled spiking LIF networks and rate networks with matched topology and input-output function. In particular there is no indication of a corresponding chaotic instability in the spiking network.

Renewal Approach to the Analysis of the Asynchronous State for Coupled Noisy Oscillators.

We develop a framework in which the activity of nonlinear pulse-coupled oscillators is posed within the renewal theory. In this approach, the evolution of the interevent density allows for a self-consistent calculation that determines the asynchronous state and its stability. This framework can readily be extended to the analysis of systems with more state variables and provides a population density treatment to evolve them in their thermodynamical limits. To demonstrate this we study a nonlinear pulse-coupled system, where couplings are dynamic and activity dependent. We investigate its stability and numerically study the nonequilibrium behavior of the system after the bifurcation. We show that this system undergoes a supercritical Hopf bifurcation to collective synchronization.

Neural representation of calling songs and their behavioral relevance in the grasshopper auditory system.

Acoustic communication plays a key role for mate attraction in grasshoppers. Males use songs to advertise themselves to females. Females evaluate the song pattern, a repetitive structure of sound syllables separated by short pauses, to recognize a conspecific male and as proxy to its fitness. In their natural habitat females often receive songs with degraded temporal structure. Perturbations may, for example, result from the overlap with other songs. We studied the response behavior of females to songs that show different signal degradations. A perturbation of an otherwise attractive song at later positions in the syllable diminished the behavioral response, whereas the same perturbation at the onset of a syllable did not affect song attractiveness. We applied naïve Bayes classifiers to the spike trains of identified neurons in the auditory pathway to explore how sensory evidence about the acoustic stimulus and its attractiveness is represented in the neuronal responses. We find that populations of three or more neurons were sufficient to reliably decode the acoustic stimulus and to predict its behavioral relevance from the single-trial integrated firing rate. A simple model of decision making simulates the female response behavior. It computes for each syllable the likelihood for the presence of an attractive song pattern as evidenced by the population firing rate. Integration across syllables allows the likelihood to reach a decision threshold and to elicit the behavioral response. The close match between model performance and animal behavior shows that a spike rate code is sufficient to enable song pattern recognition.

Cellular adaptation facilitates sparse and reliable coding in sensory pathways.

Most neurons in peripheral sensory pathways initially respond vigorously when a preferred stimulus is presented, but adapt as stimulation continues. It is unclear how this phenomenon affects stimulus coding in the later stages of sensory processing. Here, we show that a temporally sparse and reliable stimulus representation develops naturally in sequential stages of a sensory network with adapting neurons. As a modeling framework we employ a mean-field approach together with an adaptive population density treatment, accompanied by numerical simulations of spiking neural networks. We find that cellular adaptation plays a critical role in the dynamic reduction of the trial-by-trial variability of cortical spike responses by transiently suppressing self-generated fast fluctuations in the cortical balanced network. This provides an explanation for a widespread cortical phenomenon by a simple mechanism. We further show that in the insect olfactory system cellular adaptation is sufficient to explain the emergence of the temporally sparse and reliable stimulus representation in the mushroom body. Our results reveal a generic, biophysically plausible mechanism that can explain the emergence of a temporally sparse and reliable stimulus representation within a sequential processing architecture.

Adaptation reduces variability of the neuronal population code.

Sequences of events in noise-driven excitable systems with slow variables often show serial correlations among their intervals of events. Here, we employ a master equation for generalized non-renewal processes to calculate the interval and count statistics of superimposed processes governed by a slow adaptation variable. For an ensemble of neurons with spike-frequency adaptation, this results in the regularization of the population activity and an enhanced postsynaptic signal decoding. We confirm our theoretical results in a population of cortical neurons recorded in vivo.

Serial correlation in neural spike trains: experimental evidence, stochastic modeling, and single neuron variability.

The activity of spiking neurons is frequently described by renewal point process models that assume the statistical independence and identical distribution of the intervals between action potentials. However, the assumption of independent intervals must be questioned for many different types of neurons. We review experimental studies that reported the feature of a negative serial correlation of neighboring intervals, commonly observed in neurons in the sensory periphery as well as in central neurons, notably in the mammalian cortex. In our experiments we observed the same short-lived negative serial dependence of intervals in the spontaneous activity of mushroom body extrinsic neurons in the honeybee. To model serial interval correlations of arbitrary lags, we suggest a family of autoregressive point processes. Its marginal interval distribution is described by the generalized gamma model, which includes as special cases the log-normal and gamma distributions, which have been widely used to characterize regular spiking neurons. In numeric simulations we investigated how serial correlation affects the variance of the neural spike count. We show that the experimentally confirmed negative correlation reduces single-neuron variability, as quantified by the Fano factor, by up to 50%, which favors the transmission of a rate code. We argue that the feature of a negative serial correlation is likely to be common to the class of spike-frequency-adapting neurons and that it might have been largely overlooked in extracellular single-unit recordings due to spike sorting errors.