Neuron particles capture network topology and behavior from single units

While networks of neurons, glia and vascular systems enable and support brain functions, to date, mathematical tools to decode network dynamics and structure from very scarce and partially observed neuronal spiking behavior remain underdeveloped. Large neuronal networks contribute to the intrinsic neuron transfer function and observed neuronal spike trains encoding complex causal information processing, yet how this emerging causal fractal memory in the spike trains relates to the network topology is not fully understood. Towards this end, we propose a novel statistical physics inspired neuron particle model that captures the causal information flow and processing features of neuronal spiking activity. Relying on synthetic comprehensive simulations and real-world neuronal spiking activity analysis, the proposed fractional order operators governing the neuronal spiking dynamics provide insights into the memory and scale of the spike trains as well as information about the topological properties of the underlying neuronal networks. Lastly, the proposed model exhibits superior predictions of animal behavior during multiple cognitive tasks.

Models of stochastic $$\hbox {Ca}^{2+}$$ spiking

Complexity and limited knowledge render it impractical to write down the equations describing a cellular system completely. Cellular biophysics uses hypotheses-based modelling instead. How can we set up models with predictive power beyond the experimental examples used to develop them? The two textbook systems of cellular biophysics, $$\hbox {Ca}^{2+}$$ Ca 2 + signalling and neuronal membrane potential dynamics, both face this question. Both systems also have a non-equilibrium feature in common: on different time scales and for different observables, they exhibit stochastic spiking, i.e., sequences of stereotypical events that are separated by statistically distributed intervals, the interspike intervals (ISI). Here we review recent progress on the description of $$\hbox {Ca}^{2+}$$ Ca 2 + spikes in terms of blips, puffs and cellular $$\hbox {Ca}^{2+}$$ Ca 2 + spikes and focus on stochastic models that can explain the statistics of the single ISIs, in particular its mean and variance and the cell-to-cell variability of these statistics. We also review models of the stochastic integrate-and-fire type and measures like the spike-train power spectrum or the serial correlation coefficient that are used to describe neuronal spike trains. These concepts from computational neuroscience might be applicable for understanding long-term memory effects in $$\hbox {Ca}^{2+}$$ Ca 2 + spiking that extend beyond a single ISI, such as cumulative refractoriness.

A convolutional neural network for estimating synaptic connectivity from spike trains

The recent increase in reliable, simultaneous high channel count extracellular recordings is exciting for physiologists and theoreticians because it offers the possibility of reconstructing the underlying neuronal circuits. We recently presented a method of inferring this circuit connectivity from neuronal spike trains by applying the generalized linear model to cross-correlograms. Although the algorithm can do a good job of circuit reconstruction, the parameters need to be carefully tuned for each individual dataset. Here we present another method using a Convolutional Neural Network for Estimating synaptic Connectivity from spike trains. After adaptation to huge amounts of simulated data, this method robustly captures the specific feature of monosynaptic impact in a noisy cross-correlogram. There are no user-adjustable parameters. With this new method, we have constructed diagrams of neuronal circuits recorded in several cortical areas of monkeys.

Adaptation with correlated noise predicts negative interval correlations in neuronal spike trains

Negative correlations in the sequential evolution of interspike intervals (ISIs) are a signature of memory in neuronal spike-trains. They provide coding benefits including firing-rate stabilization, improved detectability of weak sensory signals, and enhanced transmission of information by improving signal-to-noise ratio. Here we predict observed ISI serial correlations from primary electrosensory afferents of weakly electric fish using an adaptive threshold model with a noisy spike threshold. We derive a general relationship between serial correlation coefficients (SCCs) and the autocorrelation function of added noise. Observed afferent spike-trains fall into two categories based on the pattern of SCCs: non-bursting units have negative SCCs which remain negative but decay to zero with increasing lags (Type I SCCs), and bursting units have oscillatory (alternating sign) SCCs which damp to zero with increasing lags (Type II SCCs). Type I SCCs are generated by low-pass filtering white noise before adding it to the spike threshold, whereas Type II SCCs are generated by high-pass filtering white noise. Thus, a single parameter (the sign of the pole of the filter) generates both types of SCCs. The filter pole (equivalently time-constant) is estimated from the observed SCCs. The predicted SCCs are in geometric progression. The theory predicts that the limiting sum of SCCs is −0.5, and this is confirmed from the expressions for the two types of filters. Observed SCCs from afferents have a limiting sum that is slightly larger at −0.475 ±0.04 (mean ±s.d.). The theoretical limit of the sum of SCCs leads to a perfect DC block in the power spectrum of the spike-train, thereby maximizing signal-to-noise ratio during signal encoding. The experimentally observed sum of SCCs is just short of this limit. We conclude by discussing the results from the perspective of optimal coding.Author summaryMany neurons spontaneously emit spikes (impulses) with a random time interval between successive spikes (interspike interval or ISI). The spike generation mechanism can have memory so that successive ISIs are dependent on one another and exhibit correlations. An ISI which is longer than the mean ISI tends to be followed by an ISI which is shorter than the mean, and vice versa. Thus, adjacent ISIs are negatively correlated, and further these correlations can extend over multiple ISIs. A simple model describing negative correlations in ISIs is an adaptive threshold with noise added to the spike threshold. A neuron becomes more resistant (refractory) to spiking immediately after a spike is output, with refractoriness increasing as more spikes are spaced closer together. Refractoriness reduces as spikes are spaced further apart. We show that a neuron can generate experimentally observed patterns of correlations by relating it to the noise in the spike threshold. Two different types of filtered noise (low-pass and high-pass) generate the observed patterns of correlations. We show that the theoretical sum of the sequence of correlations has a limiting value which maximizes the information a neuron can transmit. The observed sum of correlations is close to this limit.

Fano Factor: A Potentially Useful Information

The Fano factor, defined as the variance-to-mean ratio of spike counts in a time window, is often used to measure the variability of neuronal spike trains. However, despite its transparent definition, careless use of the Fano factor can easily lead to distorted or even wrong results. One of the problems is the unclear dependence of the Fano factor on the spiking rate, which is often neglected or handled insufficiently. In this paper we aim to explore this problem in more detail and to study the possible solution, which is to evaluate the Fano factor in the operational time. We use equilibrium renewal and Markov renewal processes as spike train models to describe the method in detail, and we provide an illustration on experimental data.

CoNNECT: Convolutional Neural Network for Estimating synaptic Connectivity from spike Trains

The recent increase in reliable, simultaneous high channel count extracellular recordings is exciting for physiologists and theoreticians, because it offers the possibility of reconstructing the underlying neuronal circuits. We recently presented a method of inferring this circuit connectivity from neuronal spike trains by applying the generalized linear model to cross-correlograms, GLMCC. Although the GLMCC algorithm can do a good job of circuit reconstruction, the parameters need to be carefully tuned for each individual dataset. Here we present another algorithm using a convolutional neural network for estimating synaptic connectivity from spike trains, CoNNECT. After adaptation to very large amounts of simulated data, this algorithm robustly captures the specific feature of monosynaptic impact in a noisy cross-correlogram. There are no user-adjustable parameters. With this new algorithm, we have constructed diagrams of neuronal circuits recorded in several cortical areas of monkeys.

Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. We review large deviations techniques useful in this context to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.