Interspike interval distributions of spiking neurons driven by fluctuating inputs

Rosetta stone for the population dynamics of spiking neuron networks

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equation governing such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

Phase-Dependent Response to Electrical Stimulation of Cortical Networks during Recurrent Epileptiform Short Discharge Generation In Vitro

The closed-loop control of pathological brain activity is a challenging task. In this study, we investigated the sensitivity of continuous epileptiform short discharge generation to electrical stimulation applied at different phases between the discharges using an in vitro 4-AP-based model of epilepsy in rat hippocampal slices. As a measure of stimulation effectiveness, we introduced a sensitivity function, which we then measured in experiments and analyzed with different biophysical and abstract mathematical models, namely, (i) the two-order subsystem of our previous Epileptor-2 model, describing short discharge generation governed by synaptic resource dynamics; (ii) a similar model governed by shunting conductance dynamics (Epileptor-2B); (iii) the stochastic leaky integrate-and-fire (LIF)-like model applied for the network; (iv) the LIF model with potassium M-channels (LIF+KM), belonging to Class II of excitability; and (v) the Epileptor-2B model with after-spike depolarization. A semi-analytic method was proposed for calculating the interspike interval (ISI) distribution and the sensitivity function in LIF and LIF+KM models, which provided parametric analysis. Sensitivity was found to increase with phase for all models except the last one. The Epileptor-2B model is favored over other models for subthreshold oscillations in the presence of large noise, based on the comparison of ISI statistics and sensitivity functions with experimental data. This study also emphasizes the stochastic nature of epileptiform discharge generation and the greater effectiveness of closed-loop stimulation in later phases of ISIs.

Silences, spikes and bursts: Three-part knot of the neural code

When a neuron breaks silence, it can emit action potentials in a number of patterns. Some responses are so sudden and intense that electrophysiologists felt the need to single them out, labelling action potentials emitted at a particularly high frequency with a metonym - bursts. Is there more to bursts than a figure of speech? After all, sudden bouts of high-frequency firing are expected to occur whenever inputs surge. The burst coding hypothesis advances that the neural code has three syllables: silences, spikes and bursts. We review evidence supporting this ternary code in terms of devoted mechanisms for burst generation, synaptic transmission and synaptic plasticity. We also review the learning and attention theories for which such a triad is beneficial.

Logarithmically scaled, gamma distributed neuronal spiking

Naturally log-scaled quantities abound in the nervous system. Distributions of these quantities have non-intuitive properties, which have implications for data analysis and the understanding of neural circuits. Here, we review the log-scaled statistics of neuronal spiking and the relevant analytical probability distributions. Recent work using log-scaling revealed that interspike intervals of forebrain neurons segregate into discrete modes reflecting spiking at different timescales and are each well-approximated by a gamma distribution. Each neuron spends most of the time in an irregular spiking 'ground state' with the longest intervals, which determines the mean firing rate of the neuron. Across the entire neuronal population, firing rates are log-scaled and well approximated by the gamma distribution, with a small number of highly active neurons and an overabundance of low rate neurons (the 'dark matter'). These results are intricately linked to a heterogeneous balanced operating regime, which confers upon neuronal circuits multiple computational advantages and has evolutionarily ancient origins.

Bursting hierarchy in an adaptive exponential integrate-and-fire network synchronization

Neuronal network synchronization has received wide interest. In the present manuscript, we study the influence of initial membrane potentials together with network topology on bursting synchronization, in particular the sequential order of stabilized bursting among neurons. We find a hierarchical phenomenon on their bursting order. With a focus on situations where network coupling advances spiking times of neurons, we grade neurons into different layers. Together with the neuronal network structure, we construct directed graphs to indicate bursting propagation between different layers. More explicitly, neurons in upper layers burst earlier than those in lower layers. More interestingly, we find that among the same layer, bursting order of neurons is mainly associated with the number of neurons they connected to the upper layer; more stimuli lead to earlier bursting. Receiving effectively the same stimuli from the upper layer, we observe neurons with fewer connections would burst earlier.

The Effects of Background Noise on a Biophysical Model of Olfactory Bulb Mitral Cells

The spiking activity of mitral cells (MC) in the olfactory bulb is a key attribute in olfactory sensory information processing to downstream cortical areas. A more detailed understanding of the modulation of MC spike statistics could shed light on mechanistic studies of olfactory bulb circuits and olfactory coding. We study the spike response of a recently developed single-compartment biophysical MC model containing seven known ionic currents and calcium dynamics subject to constant current input with background white noise. We observe rich spiking dynamics even with constant current input, including multimodal peaks in the interspike interval distribution (ISI). Although weak-to-moderate background noise for a fixed current input does not change the firing rate much, the spike dynamics can change dramatically, exhibiting non-monotonic spike variability not commonly observed in standard neuron models. We explain these dynamics with a phenomenological model of the ISI probability density function. Our study clarifies some of the complexities of MC spiking dynamics.

Stimulus presentation can enhance spiking irregularity across subcortical and cortical regions

Stimulus presentation is believed to quench neural response variability as measured by fano-factor (FF). However, the relative contributions of within-trial spike irregularity and trial-to-trial rate variability to FF fluctuations have remained elusive. Here, we introduce a principled approach for accurate estimation of spiking irregularity and rate variability in time for doubly stochastic point processes. Consistent with previous evidence, analysis showed stimulus-induced reduction in rate variability across multiple cortical and subcortical areas. However, unlike what was previously thought, spiking irregularity, was not constant in time but could be enhanced due to factors such as bursting abating the quench in the post-stimulus FF. Simulations confirmed plausibility of a time varying spiking irregularity arising from within and between pool correlations of excitatory and inhibitory neural inputs. By accurate parsing of neural variability, our approach reveals previously unnoticed changes in neural response variability and constrains candidate mechanisms that give rise to observed rate variability and spiking irregularity within brain regions.

Mounting evidence suggest neural response variability to be important for understanding and constraining the underlying neural mechanisms in a given brain area. Here, by analyzing responses across multiple brain areas and by using a principled method for parsing variability components into rate variability and spiking irregularity, we show that unlike what was previously thought, event-related quench of variability is not a brain-wide phenomenon and that point process variability and nonrenewal bursting can enhance post-stimulus spiking irregularity across certain cortical and subcortical regions. We propose possible presynaptic mechanisms that may underlie the observed heterogeneities in spiking variability across the brain.

Neural burst codes disguised as rate codes

The burst coding hypothesis posits that the occurrence of sudden high-frequency patterns of action potentials constitutes a salient syllable of the neural code. Many neurons, however, do not produce clearly demarcated bursts, an observation invoked to rule out the pervasiveness of this coding scheme across brain areas and cell types. Here we ask how detrimental ambiguous spike patterns, those that are neither clearly bursts nor isolated spikes, are for neuronal information transfer. We addressed this question using information theory and computational simulations. By quantifying how information transmission depends on firing statistics, we found that the information transmitted is not strongly influenced by the presence of clearly demarcated modes in the interspike interval distribution, a feature often used to identify the presence of burst coding. Instead, we found that neurons having unimodal interval distributions were still able to ascribe different meanings to bursts and isolated spikes. In this regime, information transmission depends on dynamical properties of the synapses as well as the length and relative frequency of bursts. Furthermore, we found that common metrics used to quantify burstiness were unable to predict the degree with which bursts could be used to carry information. Our results provide guiding principles for the implementation of coding strategies based on spike-timing patterns, and show that even unimodal firing statistics can be consistent with a bivariate neural code.

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.

Monosynaptic inference via finely-timed spikes

Observations of finely-timed spike relationships in population recordings have been used to support partial reconstruction of neural microcircuit diagrams. In this approach, fine-timescale components of paired spike train interactions are isolated and subsequently attributed to synaptic parameters. Recent perturbation studies strengthen the case for such an inference, yet the complete set of measurements needed to calibrate statistical models is unavailable. To address this gap, we study features of pairwise spiking in a large-scale in vivo dataset where presynaptic neurons were explicitly decoupled from network activity by juxtacellular stimulation. We then construct biophysical models of paired spike trains to reproduce the observed phenomenology of in vivo monosynaptic interactions, including both fine-timescale spike-spike correlations and firing irregularity. A key characteristic of these models is that the paired neurons are coupled by rapidly-fluctuating background inputs. We quantify a monosynapse's causal effect by comparing the postsynaptic train with its counterfactual, when the monosynapse is removed. Subsequently, we develop statistical techniques for estimating this causal effect from the pre- and post-synaptic spike trains. A particular focus is the justification and application of a nonparametric separation of timescale principle to implement synaptic inference. Using simulated data generated from the biophysical models, we characterize the regimes in which the estimators accurately identify the monosynaptic effect. A secondary goal is to initiate a critical exploration of neurostatistical assumptions in terms of biophysical mechanisms, particularly with regards to the challenging but arguably fundamental issue of fast, unobservable nonstationarities in background dynamics.

A Measure of Concurrent Neural Firing Activity Based on Mutual Information

Multiple methods have been developed in an attempt to quantify stimulus-induced neural coordination and to understand internal coordination of neuronal responses by examining the synchronization phenomena in neural discharge patterns. In this work we propose a novel approach to estimate the degree of concomitant firing between two neural units, based on a modified form of mutual information (MI) applied to a two-state representation of the firing activity. The binary profile of each single unit unfolds its discharge activity in time by decomposition into the state of neural quiescence/low activity and state of moderate firing/bursting. Then, the MI computed between the two binary streams is normalized by their minimum entropy and is taken as positive or negative depending on the prevalence of identical or opposite concomitant states. The resulting measure, denoted as Concurrent Firing Index based on MI (CFI_MI), relies on a single input parameter and is otherwise assumption-free and symmetric. Exhaustive validation was carried out through controlled experiments in three simulation scenarios, showing that CFI_MI is independent on firing rate and recording duration, and is sensitive to correlated and anti-correlated firing patterns. Its ability to detect non-correlated activity was assessed using ad-hoc surrogate data. Moreover, the evaluation of CFI_MI on experimental recordings of spiking activity in retinal ganglion cells brought insights into the changes of neural synchrony over time. The proposed measure offers a novel perspective on the estimation of neural synchrony, providing information on the co-occurrence of firing states in the two analyzed trains over longer temporal scales compared to existing measures.

Malleability of gamma rhythms enhances population-level correlations

An important problem in systems neuroscience is to understand how information is communicated among brain regions, and it has been proposed that communication is mediated by neuronal oscillations, such as rhythms in the gamma band. We sought to investigate this idea by using a network model with two components, a source (sending) and a target (receiving) component, both built to resemble local populations in the cerebral cortex. To measure the effectiveness of communication, we used population-level correlations in spike times between the source and target. We found that after correcting for a response time that is independent of initial conditions, spike-time correlations between the source and target are significant, due in large measure to the alignment of firing events in their gamma rhythms. But, we also found that regular oscillations cannot produce the results observed in our model simulations of cortical neurons. Surprisingly, it is the irregularity of gamma rhythms, the absence of internal clocks, together with the malleability of these rhythms and their tendency to align with external pulses - features that are known to be present in gamma rhythms in the real cortex - that produced the results observed. These findings and the mechanistic explanations we offered are our primary results. Our secondary result is a mathematical relationship between correlations and the sizes of the samples used for their calculation. As improving technology enables recording simultaneously from increasing numbers of neurons, this relationship could be useful for interpreting results from experimental recordings.

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.

Inferring and validating mechanistic models of neural microcircuits based on spike-train data

The interpretation of neuronal spike train recordings often relies on abstract statistical models that allow for principled parameter estimation and model selection but provide only limited insights into underlying microcircuits. In contrast, mechanistic models are useful to interpret microcircuit dynamics, but are rarely quantitatively matched to experimental data due to methodological challenges. Here we present analytical methods to efficiently fit spiking circuit models to single-trial spike trains. Using derived likelihood functions, we statistically infer the mean and variance of hidden inputs, neuronal adaptation properties and connectivity for coupled integrate-and-fire neurons. Comprehensive evaluations on synthetic data, validations using ground truth in-vitro and in-vivo recordings, and comparisons with existing techniques demonstrate that parameter estimation is very accurate and efficient, even for highly subsampled networks. Our methods bridge statistical, data-driven and theoretical, model-based neurosciences at the level of spiking circuits, for the purpose of a quantitative, mechanistic interpretation of recorded neuronal population activity.

It is difficult to fit mechanistic, biophysically constrained circuit models to spike train data from in vivo extracellular recordings. Here the authors present analytical methods that enable efficient parameter estimation for integrate-and-fire circuit models and inference of the underlying connectivity structure in subsampled networks.

Neural activity of heterogeneous inhibitory spiking networks with delay

We study a network of spiking neurons with heterogeneous excitabilities connected via inhibitory delayed pulses. For globally coupled systems the increase of the inhibitory coupling reduces the number of firing neurons by following a winner-takes-all mechanism. For sufficiently large transmission delay we observe the emergence of collective oscillations in the system beyond a critical coupling value. Heterogeneity promotes neural inactivation and asynchronous dynamics and its effect can be counteracted by considering longer time delays. In sparse networks, inhibition has the counterintuitive effect of promoting neural reactivation of silent neurons for sufficiently large coupling. In this regime, current fluctuations are on one side responsible for neural firing of subthreshold neurons and on the other side for their desynchronization. Therefore, collective oscillations are present only in a limited range of coupling values, which remains finite in the thermodynamic limit. Out of this range the dynamics is asynchronous and for very large inhibition neurons display a bursting behavior alternating periods of silence with periods where they fire freely in absence of any inhibition.

Information transmission of mean and variance coding in integrate-and-fire neurons

Neurons process information by translating continuous signals into patterns of discrete spike times. An open question is how much information these spike times contain about signals which modulate either the mean or the variance of the somatic currents in neurons, as is observed experimentally. Here we calculate the exact information contained in discrete spike times about a continuous signal in both encoding strategies. We show that the information content about mean modulating signals is generally substantially larger than about variance modulating signals for biological parameters. Our analysis further reveals that higher information transmission is associated with a larger proportion of nonlinear signal encoding. Our study measures the complete information content of mean and variance coding and provides a method to determine what fraction of the total information is linearly decodable.

Using computational models to predict in vivo synaptic inputs to interneuron specific 3 (IS3) cells of CA1 hippocampus that also allow their recruitment during rhythmic states

Brain coding strategies are enabled by the balance of synaptic inputs that individual neurons receive as determined by the networks in which they reside. Inhibitory cell types contribute to brain function in distinct ways but recording from specific, inhibitory cell types during behaviour to determine their contributions is highly challenging. In particular, the in vivo activities of vasoactive intestinal peptide-expressing interneuron specific 3 (IS3) cells in the hippocampus that only target other inhibitory cells are unknown at present. We perform a massive, computational exploration of possible synaptic inputs to IS3 cells using multi-compartment models and optimized synaptic parameters. We find that asynchronous, in vivo-like states that are sensitive to additional theta-timed inputs (8 Hz) exist when excitatory and inhibitory synaptic conductances are approximately equally balanced and with low numbers of activated synapses receiving correlated inputs. Specifically, under these balanced conditions, the input resistance is larger with higher mean spike firing rates relative to other activated synaptic conditions investigated. Incoming theta-timed inputs result in strongly increased spectral power relative to baseline. Thus, using a generally applicable computational approach we predict the existence and features of background, balanced states in hippocampal circuits.

Transmission of temporally correlated spike trains through synapses with short-term depression

Short-term synaptic depression, caused by depletion of releasable neurotransmitter, modulates the strength of neuronal connections in a history-dependent manner. Quantifying the statistics of synaptic transmission requires stochastic models that link probabilistic neurotransmitter release with presynaptic spike-train statistics. Common approaches are to model the presynaptic spike train as either regular or a memory-less Poisson process: few analytical results are available that describe depressing synapses when the afferent spike train has more complex, temporally correlated statistics such as bursts. Here we present a series of analytical results—from vesicle release-site occupancy statistics, via neurotransmitter release, to the post-synaptic voltage mean and variance—for depressing synapses driven by correlated presynaptic spike trains. The class of presynaptic drive considered is that fully characterised by the inter-spike-interval distribution and encompasses a broad range of models used for neuronal circuit and network analyses, such as integrate-and-fire models with a complete post-spike reset and receiving sufficiently short-time correlated drive. We further demonstrate that the derived post-synaptic voltage mean and variance allow for a simple and accurate approximation of the firing rate of the post-synaptic neuron, using the exponential integrate-and-fire model as an example. These results extend the level of biological detail included in models of synaptic transmission and will allow for the incorporation of more complex and physiologically relevant firing patterns into future studies of neuronal networks.

Synapses between neurons transmit signals with strengths that vary with the history of their activity, over scales from milliseconds to decades. Short-term changes in synaptic strength modulate and sculpt ongoing neuronal activity, whereas long-term changes underpin memory formation. Here we focus on changes of strength over timescales of less than a second caused by transitory depletion of the neurotransmitters that carry signals across the synapse. Neurotransmitters are stored in small vesicles that release their contents, with a certain probability, when the presynaptic neuron is active. Once a vesicle has been used it is replenished after a variable delay. There is therefore a complex interaction between the pattern of incoming signals to the synapse and the probablistic release and restock of packaged neurotransmitter. Here we extend existing models to examine how correlated synaptic activity is transmitted through synapses and affects the voltage fluctuations and firing rate of the target neuron. Our results provide a framework that will allow for the inclusion of biophysically realistic synaptic behaviour in studies of neuronal circuits.

Colored noise and a stochastic fractional model for correlated inputs and adaptation in neuronal firing

High variability in the neuronal response to stimulations and the adaptation phenomenon cannot be explained by the standard stochastic leaky integrate-and-fire model. The main reason is that the uncorrelated inputs involved in the model are not realistic. There exists some form of dependency between the inputs, and it can be interpreted as memory effects. In order to include these physiological features in the standard model, we reconsider it with time-dependent coefficients and correlated inputs. Due to its hard mathematical tractability, we perform simulations of it for a wide investigation of its output. A Gauss-Markov process is constructed for approximating its non-Markovian dynamics. The first passage time probability density of such a process can be numerically evaluated, and it can be used to fit the histograms of simulated firing times. Some estimates of the moments of firing times are also provided. The effect of the correlation time of the inputs on firing densities and on firing rates is shown. An exponential probability density of the first firing time is estimated for low values of input current and high values of correlation time. For comparison, a simulation-based investigation is also carried out for a fractional stochastic model that allows to preserve the memory of the time evolution of the neuronal membrane potential. In this case, the memory parameter that affects the firing activity is the fractional derivative order. In both models an adaptation level of spike frequency is attained, even if along different modalities. Comparisons and discussion of the obtained results are provided.

Physiological models of the lateral superior olive

In computational biology, modeling is a fundamental tool for formulating, analyzing and predicting complex phenomena. Most neuron models, however, are designed to reproduce certain small sets of empirical data. Hence their outcome is usually not compatible or comparable with other models or datasets, making it unclear how widely applicable such models are. In this study, we investigate these aspects of modeling, namely credibility and generalizability, with a specific focus on auditory neurons involved in the localization of sound sources. The primary cues for binaural sound localization are comprised of interaural time and level differences (ITD/ILD), which are the timing and intensity differences of the sound waves arriving at the two ears. The lateral superior olive (LSO) in the auditory brainstem is one of the locations where such acoustic information is first computed. An LSO neuron receives temporally structured excitatory and inhibitory synaptic inputs that are driven by ipsi- and contralateral sound stimuli, respectively, and changes its spike rate according to binaural acoustic differences. Here we examine seven contemporary models of LSO neurons with different levels of biophysical complexity, from predominantly functional ones (‘shot-noise’ models) to those with more detailed physiological components (variations of integrate-and-fire and Hodgkin-Huxley-type). These models, calibrated to reproduce known monaural and binaural characteristics of LSO, generate largely similar results to each other in simulating ITD and ILD coding. Our comparisons of physiological detail, computational efficiency, predictive performances, and further expandability of the models demonstrate (1) that the simplistic, functional LSO models are suitable for applications where low computational costs and mathematical transparency are needed, (2) that more complex models with detailed membrane potential dynamics are necessary for simulation studies where sub-neuronal nonlinear processes play important roles, and (3) that, for general purposes, intermediate models might be a reasonable compromise between simplicity and biological plausibility.

Computational models help our understanding of complex biological systems, by identifying their key elements and revealing their operational principles. Close comparisons between model predictions and empirical observations ensure our confidence in a model as a building block for further applications. Most current neuronal models, however, are constructed to replicate only a small specific set of experimental data. Thus, it is usually unclear how these models can be generalized to different datasets and how they compare with each other. In this paper, seven neuronal models are examined that are designed to reproduce known physiological characteristics of auditory neurons involved in the detection of sound source location. Despite their different levels of complexity, the models generate largely similar results when their parameters are tuned with common criteria. Comparisons show that simple models are computationally more efficient and theoretically transparent, and therefore suitable for rigorous mathematical analyses and engineering applications including real-time simulations. In contrast, complex models are necessary for investigating the relationship between underlying biophysical processes and sub- and suprathreshold spiking properties, although they have a large number of unconstrained, unverified parameters. Having identified their advantages and drawbacks, these auditory neuron models may readily be used for future studies and applications.

How does transient signaling input affect the spike timing of postsynaptic neuron near the threshold regime: an analytical study

The noisy threshold regime, where even a small set of presynaptic neurons can significantly affect postsynaptic spike-timing, is suggested as a key requisite for computation in neurons with high variability. It also has been proposed that signals under the noisy conditions are successfully transferred by a few strong synapses and/or by an assembly of nearly synchronous synaptic activities. We analytically investigate the impact of a transient signaling input on a leaky integrate-and-fire postsynaptic neuron that receives background noise near the threshold regime. The signaling input models a single strong synapse or a set of synchronous synapses, while the background noise represents a lot of weak synapses. We find an analytic solution that explains how the first-passage time (ISI) density is changed by transient signaling input. The analysis allows us to connect properties of the signaling input like spike timing and amplitude with postsynaptic first-passage time density in a noisy environment. Based on the analytic solution, we calculate the Fisher information with respect to the signaling input’s amplitude. For a wide range of amplitudes, we observe a non-monotonic behavior for the Fisher information as a function of background noise. Moreover, Fisher information non-trivially depends on the signaling input’s amplitude; changing the amplitude, we observe one maximum in the high level of the background noise. The single maximum splits into two maximums in the low noise regime. This finding demonstrates the benefit of the analytic solution in investigating signal transfer by neurons.

Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models.

Characterizing the dynamics of biophysically modeled, large neuronal networks usually involves extensive numerical simulations. As an alternative to this expensive procedure we propose efficient models that describe the network activity in terms of a few ordinary differential equations. These systems are simple to solve and allow for convenient investigations of asynchronous, oscillatory or chaotic network states because linear stability analyses and powerful related methods are readily applicable. We build upon two research lines on which substantial efforts have been exerted in the last two decades: (i) the development of single neuron models of reduced complexity that can accurately reproduce a large repertoire of observed neuronal behavior, and (ii) different approaches to approximate the Fokker-Planck equation that represents the collective dynamics of large neuronal networks. We combine these advances and extend recent approximation methods of the latter kind to obtain spike rate models that surprisingly well reproduce the macroscopic dynamics of the underlying neuronal network. At the same time the microscopic properties are retained through the single neuron model parameters. To enable a fast adoption we have released an efficient Python implementation as open source software under a free license.

Intrinsically-generated fluctuating activity in excitatory-inhibitory networks

Recurrent networks of non-linear units display a variety of dynamical regimes depending on the structure of their synaptic connectivity. A particularly remarkable phenomenon is the appearance of strongly fluctuating, chaotic activity in networks of deterministic, but randomly connected rate units. How this type of intrinsically generated fluctuations appears in more realistic networks of spiking neurons has been a long standing question. To ease the comparison between rate and spiking networks, recent works investigated the dynamical regimes of randomly-connected rate networks with segregated excitatory and inhibitory populations, and firing rates constrained to be positive. These works derived general dynamical mean field (DMF) equations describing the fluctuating dynamics, but solved these equations only in the case of purely inhibitory networks. Using a simplified excitatory-inhibitory architecture in which DMF equations are more easily tractable, here we show that the presence of excitation qualitatively modifies the fluctuating activity compared to purely inhibitory networks. In presence of excitation, intrinsically generated fluctuations induce a strong increase in mean firing rates, a phenomenon that is much weaker in purely inhibitory networks. Excitation moreover induces two different fluctuating regimes: for moderate overall coupling, recurrent inhibition is sufficient to stabilize fluctuations; for strong coupling, firing rates are stabilized solely by the upper bound imposed on activity, even if inhibition is stronger than excitation. These results extend to more general network architectures, and to rate networks receiving noisy inputs mimicking spiking activity. Finally, we show that signatures of the second dynamical regime appear in networks of integrate-and-fire neurons.

Solving the two-dimensional Fokker-Planck equation for strongly correlated neurons

Pairs of neurons in brain networks often share much of the input they receive from other neurons. Due to essential nonlinearities of the neuronal dynamics, the consequences for the correlation of the output spike trains are generally not well understood. Here we analyze the case of two leaky integrate-and-fire neurons using an approach which is nonperturbative with respect to the degree of input correlation. Our treatment covers both weakly and strongly correlated dynamics, generalizing previous results based on linear response theory.

Lognormal firing rate distribution reveals prominent fluctuation-driven regime in spinal motor networks

When spinal circuits generate rhythmic movements it is important that the neuronal activity remains within stable bounds to avoid saturation and to preserve responsiveness. Here, we simultaneously record from hundreds of neurons in lumbar spinal circuits of turtles and establish the neuronal fraction that operates within either a 'mean-driven' or a 'fluctuation-driven' regime. Fluctuation-driven neurons have a 'supralinear' input-output curve, which enhances sensitivity, whereas the mean-driven regime reduces sensitivity. We find a rich diversity of firing rates across the neuronal population as reflected in a lognormal distribution and demonstrate that half of the neurons spend at least 50 % of the time in the 'fluctuation-driven' regime regardless of behavior. Because of the disparity in input-output properties for these two regimes, this fraction may reflect a fine trade-off between stability and sensitivity in order to maintain flexibility across behaviors.

Burst Firing Enhances Neural Output Correlation

Neurons communicate and transmit information predominantly through spikes. Given that experimentally observed neural spike trains in a variety of brain areas can be highly correlated, it is important to investigate how neurons process correlated inputs. Most previous work in this area studied the problem of correlation transfer analytically by making significant simplifications on neural dynamics. Temporal correlation between inputs that arises from synaptic filtering, for instance, is often ignored when assuming that an input spike can at most generate one output spike. Through numerical simulations of a pair of leaky integrate-and-fire (LIF) neurons receiving correlated inputs, we demonstrate that neurons in the presence of synaptic filtering by slow synapses exhibit strong output correlations. We then show that burst firing plays a central role in enhancing output correlations, which can explain the above-mentioned observation because synaptic filtering induces bursting. The observed changes of correlations are mostly on a long time scale. Our results suggest that other features affecting the prevalence of neural burst firing in biological neurons, e.g., adaptive spiking mechanisms, may play an important role in modulating the overall level of correlations in neural networks.

Time-invariant feed-forward inhibition of Purkinje cells in the cerebellar cortex in vivo

KEY POINTS

We performed extracellular recording of pairs of interneuron-Purkinje cells in vivo. A single interneuron produces a substantial, short-lasting, inhibition of Purkinje cells. Feed-forward inhibition is associated with characteristic asymmetric cross-correlograms. In vivo, Purkinje cell spikes only depend on the most recent synaptic activity.

ABSTRACT

Cerebellar molecular layer interneurons are considered to control the firing rate and spike timing of Purkinje cells. However, interactions between these cell types are largely unexplored in vivo. Using tetrodes, we performed simultaneous extracellular recordings of neighbouring Purkinje cells and molecular layer interneurons, presumably basket cells, in adult rats in vivo. The high levels of afferent synaptic activity encountered in vivo yield irregular spiking and reveal discharge patterns characteristic of feed-forward inhibition, thus suggesting an overlap of the afferent excitatory inputs between Purkinje cells and basket cells. Under conditions of intense background synaptic inputs, interneuron spikes exert a short-lasting inhibitory effect, delaying the following Purkinje cell spike by an amount remarkably independent of the Purkinje cell firing cycle. This effect can be explained by the short memory time of the Purkinje cell potential as a result of the intense incoming synaptic activity. Finally, we found little evidence for any involvement of the interneurons that we recorded with the cerebellar high-frequency oscillations promoting Purkinje cell synchrony. The rapid interactions between interneurons and Purkinje cells might be of particular importance in fine motor control because the inhibitory action of interneurons on Purkinje cells leads to deep cerebellar nuclear disinhibition and hence increased cerebellar output.

Anomalous neuronal responses to fluctuated inputs

The irregular firing of a cortical neuron is thought to result from a highly fluctuating drive that is generated by the balance of excitatory and inhibitory synaptic inputs. A previous study reported anomalous responses of the Hodgkin-Huxley neuron to the fluctuated inputs where an irregularity of spike trains is inversely proportional to an input irregularity. In the current study, we investigated the origin of these anomalous responses with the Hindmarsh-Rose neuron model, map-based models, and a simple mixture of interspike interval distributions. First, we specified the parameter regions for the bifurcations in the Hindmarsh-Rose model, and we confirmed that the model reproduced the anomalous responses in the dynamics of the saddle-node and subcritical Hopf bifurcations. For both bifurcations, the Hindmarsh-Rose model shows bistability in the resting state and the repetitive firing state, which indicated that the bistability was the origin of the anomalous input-output relationship. Similarly, the map-based model that contained bistability reproduced the anomalous responses, while the model without bistability did not. These results were supported by additional findings that the anomalous responses were reproduced by mimicking the bistable firing with a mixture of two different interspike interval distributions. Decorrelation of spike trains is important for neural information processing. For such spike train decorrelation, irregular firing is key. Our results indicated that irregular firing can emerge from fluctuating drives, even weak ones, under conditions involving bistability. The anomalous responses, therefore, contribute to efficient processing in the brain.

A reconfigurable visual-programming library for real-time closed-loop cellular electrophysiology

Most of the software platforms for cellular electrophysiology are limited in terms of flexibility, hardware support, ease of use, or re-configuration and adaptation for non-expert users. Moreover, advanced experimental protocols requiring real-time closed-loop operation to investigate excitability, plasticity, dynamics, are largely inaccessible to users without moderate to substantial computer proficiency. Here we present an approach based on MATLAB/Simulink, exploiting the benefits of LEGO-like visual programming and configuration, combined to a small, but easily extendible library of functional software components. We provide and validate several examples, implementing conventional and more sophisticated experimental protocols such as dynamic-clamp or the combined use of intracellular and extracellular methods, involving closed-loop real-time control. The functionality of each of these examples is demonstrated with relevant experiments. These can be used as a starting point to create and support a larger variety of electrophysiological tools and methods, hopefully extending the range of default techniques and protocols currently employed in experimental labs across the world.

Stein's neuronal model with pooled renewal input

The input of Stein's model of a single neuron is usually described by using a Poisson process, which is assumed to represent the behaviour of spikes pooled from a large number of presynaptic spike trains. However, such a description of the input is not always appropriate as the variability cannot be separated from the intensity. Therefore, we create and study Stein's model with a more general input, a sum of equilibrium renewal processes. The mean and variance of the membrane potential are derived for this model. Using these formulas and numerical simulations, the model is analyzed to study the influence of the input variability on the properties of the membrane potential and the output spike trains. The generalized Stein's model is compared with the original Stein's model with Poissonian input using the relative difference of variances of membrane potential at steady state and the integral square error of output interspike intervals. Both of the criteria show large differences between the models for input with high variability.

Analytical approximations of the firing rate of an adaptive exponential integrate-and-fire neuron in the presence of synaptic noise

Computational models offer a unique tool for understanding the network-dynamical mechanisms which mediate between physiological and biophysical properties, and behavioral function. A traditional challenge in computational neuroscience is, however, that simple neuronal models which can be studied analytically fail to reproduce the diversity of electrophysiological behaviors seen in real neurons, while detailed neuronal models which do reproduce such diversity are intractable analytically and computationally expensive. A number of intermediate models have been proposed whose aim is to capture the diversity of firing behaviors and spike times of real neurons while entailing the simplest possible mathematical description. One such model is the exponential integrate-and-fire neuron with spike rate adaptation (aEIF) which consists of two differential equations for the membrane potential (V) and an adaptation current (w). Despite its simplicity, it can reproduce a wide variety of physiologically observed spiking patterns, can be fit to physiological recordings quantitatively, and, once done so, is able to predict spike times on traces not used for model fitting. Here we compute the steady-state firing rate of aEIF in the presence of Gaussian synaptic noise, using two approaches. The first approach is based on the 2-dimensional Fokker-Planck equation that describes the (V,w)-probability distribution, which is solved using an expansion in the ratio between the time constants of the two variables. The second is based on the firing rate of the EIF model, which is averaged over the distribution of the w variable. These analytically derived closed-form expressions were tested on simulations from a large variety of model cells quantitatively fitted to in vitro electrophysiological recordings from pyramidal cells and interneurons. Theoretical predictions closely agreed with the firing rate of the simulated cells fed with in-vivo-like synaptic noise.

Switching mechanisms and bout times in a pair of reciprocally inhibitory neurons

Within the appropriate parameter regime, a deterministic model of a pair of mutually inhibitory neurons receiving excitatory driving currents exhibits bistability-each of the two stable states corresponds to one neuron being active and the other being quiescent. The presence of noise in the driving currents results in a system that randomly switches back and forth between these two states, causing alternating bouts of spiking activity. In this work, we examine the random bout durations of the two neurons and dependence on system parameters. We find that bout durations of each neuron are exponentially distributed, with changes in system parameters altering only the mean of the distribution. Synaptic inhibition independently controls the bout durations of the two neurons-the mean bout time of a neuron is a function of efferent (or outgoing) inhibition, and is independent of afferent (or incoming) inhibition. Furthermore, we find that the mean bout time of a neuron exhibits a critical dependence on the time course (rather than amplitude) of efferent inhibition-mean bout time of a neuron grows exponentially with the time course of efferent inhibition, and the growth rate of this exponential function depends only on the excitatory driving current to that neuron (and not on any other system parameters). We discuss the relevance of our results to the regulation of sleep-wake cycling by medullary and pontine structures within the brain.

Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons

Asynchronous activity in balanced networks of excitatory and inhibitory neurons is believed to constitute the primary medium for the propagation and transformation of information in the neocortex. Here we show that an unstructured, sparsely connected network of model spiking neurons can display two fundamentally different types of asynchronous activity that imply vastly different computational properties. For weak synaptic couplings, the network at rest is in the well-studied asynchronous state, in which individual neurons fire irregularly at constant rates. In this state, an external input leads to a highly redundant response of different neurons that favors information transmission but hinders more complex computations. For strong couplings, we find that the network at rest displays rich internal dynamics, in which the firing rates of individual neurons fluctuate strongly in time and across neurons. In this regime, the internal dynamics interact with incoming stimuli to provide a substrate for complex information processing and learning.

A complex-valued firing-rate model that approximates the dynamics of spiking networks

Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.

How adaptation currents change threshold, gain, and variability of neuronal spiking

Many types of neurons exhibit spike rate adaptation, mediated by intrinsic slow K(+) currents, which effectively inhibit neuronal responses. How these adaptation currents change the relationship between in vivo like fluctuating synaptic input, spike rate output, and the spike train statistics, however, is not well understood. In this computational study we show that an adaptation current that primarily depends on the subthreshold membrane voltage changes the neuronal input-output relationship (I-O curve) subtractively, thereby increasing the response threshold, and decreases its slope (response gain) for low spike rates. A spike-dependent adaptation current alters the I-O curve divisively, thus reducing the response gain. Both types of an adaptation current naturally increase the mean interspike interval (ISI), but they can affect ISI variability in opposite ways. A subthreshold current always causes an increase of variability while a spike-triggered current decreases high variability caused by fluctuation-dominated inputs and increases low variability when the average input is large. The effects on I-O curves match those caused by synaptic inhibition in networks with asynchronous irregular activity, for which we find subtractive and divisive changes caused by external and recurrent inhibition, respectively. Synaptic inhibition, however, always increases the ISI variability. We analytically derive expressions for the I-O curve and ISI variability, which demonstrate the robustness of our results. Furthermore, we show how the biophysical parameters of slow K(+) conductances contribute to the two different types of an adaptation current and find that Ca(2+)-activated K(+) currents are effectively captured by a simple spike-dependent description, while muscarine-sensitive or Na(+)-activated K(+) currents show a dominant subthreshold component.

Information filtering by synchronous spikes in a neural population

Information about time-dependent sensory stimuli is encoded by the spike trains of neurons. Here we consider a population of uncoupled but noisy neurons (each subject to some intrinsic noise) that are driven by a common broadband signal. We ask specifically how much information is encoded in the synchronous activity of the population and how this information transfer is distributed with respect to frequency bands. In order to obtain some insight into the mechanism of information filtering effects found previously in the literature, we develop a mathematical framework to calculate the coherence of the synchronous output with the common stimulus for populations of simple neuron models. Within this frame, the synchronous activity is treated as the product of filtered versions of the spike trains of a subset of neurons. We compare our results for the simple cases of (1) a Poisson neuron with a rate modulation and (2) an LIF neuron with intrinsic white current noise and a current stimulus. For the Poisson neuron, formulas are particularly simple but show only a low-pass behavior of the coherence of synchronous activity. For the LIF model, in contrast, the coherence function of the synchronous activity shows a clear peak at high frequencies, comparable to recent experimental findings. We uncover the mechanism for this shift in the maximum of the coherence and discuss some biological implications of our findings.

Statistical properties of superimposed stationary spike trains

The Poisson process is an often employed model for the activity of neuronal populations. It is known, though, that superpositions of realistic, non- Poisson spike trains are not in general Poisson processes, not even for large numbers of superimposed processes. Here we construct superimposed spike trains from intracellular in vivo recordings from rat neocortex neurons and compare their statistics to specific point process models. The constructed superimposed spike trains reveal strong deviations from the Poisson model. We find that superpositions of model spike trains that take the effective refractoriness of the neurons into account yield a much better description. A minimal model of this kind is the Poisson process with dead-time (PPD). For this process, and for superpositions thereof, we obtain analytical expressions for some second-order statistical quantities—like the count variability, inter-spike interval (ISI) variability and ISI correlations—and demonstrate the match with the in vivo data. We conclude that effective refractoriness is the key property that shapes the statistical properties of the superposition spike trains. We present new, efficient algorithms to generate superpositions of PPDs and of gamma processes that can be used to provide more realistic background input in simulations of networks of spiking neurons. Using these generators, we show in simulations that neurons which receive superimposed spike trains as input are highly sensitive for the statistical effects induced by neuronal refractoriness.