Exponential Integrate-and-fire Research Articles

Synchronization, Stochasticity, and Phase Waves in Neuronal Networks With Spatially-Structured Connectivity.

Oscillations in the beta/low gamma range (10–45 Hz) are recorded in diverse neural structures. They have successfully been modeled as sparsely synchronized oscillations arising from reciprocal interactions between randomly connected excitatory (E) pyramidal cells and local interneurons (I). The synchronization of spatially distant oscillatory spiking E–I modules has been well-studied in the rate model framework but less so for modules of spiking neurons. Here, we first show that previously proposed modifications of rate models provide a quantitative description of spiking E–I modules of Exponential Integrate-and-Fire (EIF) neurons. This allows us to analyze the dynamical regimes of sparsely synchronized oscillatory E–I modules connected by long-range excitatory interactions, for two modules, as well as for a chain of such modules. For modules with a large number of neurons (> 105), we obtain results similar to previously obtained ones based on the classic deterministic Wilson-Cowan rate model, with the added bonus that the results quantitatively describe simulations of spiking EIF neurons. However, for modules with a moderate (~ 104) number of neurons, stochastic variations in the spike emission of neurons are important and need to be taken into account. On the one hand, they modify the oscillations in a way that tends to promote synchronization between different modules. On the other hand, independent fluctuations on different modules tend to disrupt synchronization. The correlations between distant oscillatory modules can be described by stochastic equations for the oscillator phases that have been intensely studied in other contexts. On shorter distances, we develop a description that also takes into account amplitude modes and that quantitatively accounts for our simulation data. Stochastic dephasing of neighboring modules produces transient phase gradients and the transient appearance of phase waves. We propose that these stochastically-induced phase waves provide an explanative framework for the observations of traveling waves in the cortex during beta oscillations.

Controlling Complexity of Cerebral Cortex Simulations-II: Streamlined Microcircuits.

Recently, Markram et al. (2015) presented a model of the rat somatosensory microcircuit (Markram model). Their model is high in anatomical and physiological detail, and its simulation requires supercomputers. The lack of neuroinformatics and computing power is an obstacle for using a similar approach to build models of other cortical areas or larger cortical systems. Simplified neuron models offer an attractive alternative to high-fidelity Hodgkin-Huxley-type neuron models, but their validity in modeling cortical circuits is unclear. We simplified the Markram model to a network of exponential integrate-and-fire (EIF) neurons that runs on a single CPU core in reasonable time. We analyzed the electrophysiology and the morphology of the Markram model neurons with eFel and NeuroM tools, provided by the Blue Brain Project. We then constructed neurons with few compartments and averaged parameters from the reference model. We used the CxSystem simulation framework to explore the role of short-term plasticity and GABA and NMDA synaptic conductances in replicating oscillatory phenomena in the Markram model. We show that having a slow inhibitory synaptic conductance (GABA allows replication of oscillatory behavior in the high-calcium state. Furthermore, we show that qualitatively similar dynamics are seen even with a reduced number of cell types (from 55 to 17 types). This reduction halved the computation time. Our results suggest that qualitative dynamics of cortical microcircuits can be studied using limited neuroinformatics and computing resources supporting parameter exploration and simulation of cortical systems. The simplification procedure can easily be adapted to studying other microcircuits for which sparse electrophysiological and morphological data are available.

An Efficient Population Density Method for Modeling Neural Networks with Synaptic Dynamics Manifesting Finite Relaxation Time and Short-Term Plasticity.

When incorporating more realistic synaptic dynamics, the computational efficiency of population density methods (PDMs) declines sharply due to the increase in the dimension of master equations. To avoid such a decline, we develop an efficient PDM, termed colored-synapse PDM (csPDM), in which the dimension of the master equations does not depend on the number of synapse-associated state variables in the underlying network model. Our goal is to allow the PDM to incorporate realistic synaptic dynamics that possesses not only finite relaxation time but also short-term plasticity (STP). The model equations of csPDM are derived based on the diffusion approximation on synaptic dynamics and probability density function methods for Langevin equations with colored noise. Numerical examples, given by simulations of the population dynamics of uncoupled exponential integrate-and-fire (EIF) neurons, show good agreement between the results of csPDM and Monte Carlo simulations (MCSs). Compared to the original full-dimensional PDM (fdPDM), the csPDM reveals more excellent computational efficiency because of the lower dimension of the master equations. In addition, it permits network dynamics to possess the short-term plastic characteristics inherited from plastic synapses. The novel csPDM has potential applicability to any spiking neuron models because of no assumptions on neuronal dynamics, and, more importantly, this is the first report of PDM to successfully encompass short-term facilitation/depression properties.

Random Sampling with Interspike-Intervals of the Exponential Integrate and Fire Neuron: A Computational Interpretation of UP-States.

Oscillations between high and low values of the membrane potential (UP and DOWN states respectively) are an ubiquitous feature of cortical neurons during slow wave sleep and anesthesia. Nevertheless, a surprisingly small number of quantitative studies have been conducted only that deal with this phenomenon’s implications for computation.Here we present a novel theory that explains on a detailed mathematical level the computational benefits of UP states. The theory is based on random sampling by means of interspike intervals (ISIs) of the exponential integrate and fire (EIF) model neuron, such that each spike is considered a sample, whose analog value corresponds to the spike’s preceding ISI. As we show, the EIF’s exponential sodium current, that kicks in when balancing a noisy membrane potential around values close to the firing threshold, leads to a particularly simple, approximative relationship between the neuron’s ISI distribution and input current. Approximation quality depends on the frequency spectrum of the current and is improved upon increasing the voltage baseline towards threshold. Thus, the conceptually simpler leaky integrate and fire neuron that is missing such an additional current boost performs consistently worse than the EIF and does not improve when voltage baseline is increased. For the EIF in contrast, the presented mechanism is particularly effective in the high-conductance regime, which is a hallmark feature of UP-states.Our theoretical results are confirmed by accompanying simulations, which were conducted for input currents of varying spectral composition. Moreover, we provide analytical estimations of the range of ISI distributions the EIF neuron can sample from at a given approximation level. Such samples may be considered by any algorithmic procedure that is based on random sampling, such as Markov Chain Monte Carlo or message-passing methods.Finally, we explain how spike-based random sampling relates to existing computational theories about UP states during slow wave sleep and present possible extensions of the model in the context of spike-frequency adaptation.

Non-selective excitatory feedback and precise spike timing produce selective relative inhibition

The essence of feedback is key to understand the brain. However, inconsistencies exist between different observation levels. In neurophysiology, feedback connections are considered as excitatory. On the other hand, several electrophysiology and neuroimaging studies suggest that feedback connections are inhibitory. Additionally, many computational theories (such as predictive coding) assume that feedback produces selective inhibition of a subset of inputs; this often requires dedicated feedback circuitry with inhibitory connections that mirror the feed-forward excitatory connectivity. Here, we show that relative inhibition of a selected subset of inputs can be produced by strictly excitatory and non-selective feedback connections, together with precise spike times. As shown in Figure 1.A, we designed a simplified neural network with one high-level excitatory neuron and two low-level excitatory neurons. Each neuron, modelled by a leaky integrate-and-fire (LIF) unit or an exponential integrate-and-fire (EIF) unit, represents a distinct neuron population. The two low-level neurons have the same constant input (with additive noise). The high-level neuron (A) is “selective” for inputs from low-level neuron (B) (strongly weighted feed-forward connection) while discounting inputs from low-level neuron (C) (weakly weighted connection). In turn, the high-level neuron (A) sends equivalent (non-selective) excitatory feedback to both low-level neurons. The conduction time of both feedforward and feedback connections is denoted by τ. Figure 1 A. The neural network model B. Results of parameter exploring with different conduction delays τ. Positive values indicate selective inhibition of neuron B relative to neuron C. C. Difference in advancement time (a measure of feedback excitation) ... We hypothesized that precise spike timing relations between the 3 neurons would result in feedback inhibition of low-level neuron (B) relative to low-level neuron (C). That is, feedback selectively inhibits the strongly-weighted inputs of the high-level neuron. To quantify this effect, we measured the increase of firing rate of neuron (C) relative to (B), for various values of the following parameters: neuron model types (LIF or EIF), membrane time constant, intensity of the constant input to low-level neurons, additive noise range to low-level neurons, and conduction delay τ. Results for parameter τ are shown in Figure 1.B as an example. All results indicate that it is possible to generate a relative inhibitory effect with biologically plausible parameters. The principle behind this effect can be simply summarized as a sensitivity difference to incoming (feedback) spikes at different stages of the leaky integrate-and-fire process (as observed experimentally in neuronal recordings [1]). As shown in Figure 1.C, the effect on a low-level neuron of receiving a feedback spike changes dramatically according to the feedback time relative to the preceding output spike from the low-level neuron. When the time delay between the output spike and the feedback spike is short (which is more likely for low-level neuron B due to its strong connectivity to the higher-level neuron), excitatory feedback has relatively little influence on the low-level neuron’s firing, because it arrives at a time when the neuron is too far from threshold; the leak will then reduce any contribution of the feedback spike before threshold is reached. This, however, is less likely to occur for neuron C, whose output spikes are decorrelated from feedback spikes. As a result, relative feedback inhibition emerges selectively for the strongly-weighted low-level neurons (as postulated by predictive coding theory).

Dynamics of the exponential integrate-and-fire model with slow currents and adaptation

In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.

Non-instantaneous synaptic transmission in spiking neuron networks and equivalence with delay distribution

Synapses in neuron networks filter incoming spikes with a wide variety of time constants, affecting the stability of various collective dynamics [1], the selectivity in transmitting information [2] and the reactivity to suddenly appearing exogenous stimuli [3]. Even for extremely simplified models of spiking neurons and network connectivities, theoretical approaches including non-instantaneous transmission rely on approximations valid for relatively small time scales [3,4] or on the assumption of a quasi-adiabatic dynamical regime [5]. Here, we addressed this issue working out a unified framework in which from small to large synaptic time scales the same approximated theoretical description holds for the firing rate dynamics of spiking neuron networks. Starting from single-neuron stochastic dynamics, we derived a set of ordinary differential equations for the population emission rate relying on the spectral expansion of the associated Fokker-Planck equation, eventually extending to the colored noise case a previous derivation obtained under white noise hypothesis [6]. The resulting population dynamics valid under a low-noise approximation, took into account also finite-size effects. This allowed to compare theoretical power spectra of population emission rates with those estimated from extensive network simulations performed with NEST [7]. Match between theory and simulation was investigated for different integrate-and-fire (IF) neuron models with different levels of complexity, e.g., VLSI IF (VIF [6], a generalization of the Perfect IF model), Leaky IF (LIF), and Exponential IF (EIF), confirming the effectiveness of the reduced theoretical description. Effectiveness which in turn could be further improved relying on higher-order perturbative terms in the fluctuation size of the synaptic current for any synaptic time scale. Moreover, such perturbative approach allowed to avoid the analytical difficulties of dealing with multi-dimensional Fokker-Planck equations with discontinuous boundary conditions. A simplification which helped in highlighting as main result that networks of neurons with non-instantaneous synapses can, under fairly broad assumptions, be described by equivalent networks with suitable distributions of transmission delays. A formal analogy valid from simple VIF models to more realistic point-like simplifications like LIF or EIF neurons.

From spiking neuron models to linear-nonlinear models.

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.

Adaptive exponential integrate-and-fire model

Adaptive exponential integrate-and-fire model

How Noise Affects the Synchronization Properties of Recurrent Networks of Inhibitory Neurons

GABAergic interneurons play a major role in the emergence of various types of synchronous oscillatory patterns of activity in the central nervous system. Motivated by these experimental facts, modeling studies have investigated mechanisms for the emergence of coherent activity in networks of inhibitory neurons. However, most of these studies have focused either when the noise in the network is absent or weak or in the opposite situation when it is strong. Hence, a full picture of how noise affects the dynamics of such systems is still lacking. The aim of this letter is to provide a more comprehensive understanding of the mechanisms by which the asynchronous states in large, fully connected networks of inhibitory neurons are destabilized as a function of the noise level. Three types of single neuron models are considered: the leaky integrate-and-fire (LIF) model, the exponential integrate-and-fire (EIF), model and conductance-based models involving sodium and potassium Hodgkin-Huxley (HH) currents. We show that in all models, the instabilities of the asynchronous state can be classified in two classes. The first one consists of clustering instabilities, which exist in a restricted range of noise. These instabilities lead to synchronous patterns in which the population of neurons is broken into clusters of synchronously firing neurons. The irregularity of the firing patterns of the neurons is weak. The second class of instabilities, termed oscillatory firing rate instabilities, exists at any value of noise. They lead to cluster state at low noise. As the noise is increased, the instability occurs at larger coupling, and the pattern of firing that emerges becomes more irregular. In the regime of high noise and strong coupling, these instabilities lead to stochastic oscillations in which neurons fire in an approximately Poisson way with a common instantaneous probability of firing that oscillates in time.

How Noise Affects the Synchronization Properties of Recurrent Networks of Inhibitory Neurons

GABAergic interneurons play a major role in the emergence of various types of synchronous oscillatory patterns of activity in the central nervous system. Motivated by these experimental facts, modeling studies have investigated mechanisms for the emergence of coherent activity in networks of inhibitory neurons. However, most of these studies have focused either when the noise in the network is absent or weak or in the opposite situation when it is strong. Hence, a full picture of how noise affects the dynamics of such systems is still lacking. The aim of this letter is to provide a more comprehensive understanding of the mechanisms by which the asynchronous states in large, fully connected networks of inhibitory neurons are destabilized as a function of the noise level. Three types of single neuron models are considered: the leaky integrate-and-fire (LIF) model, the exponential integrate-and-fire (EIF), model and conductance-based models involving sodium and potassium Hodgkin-Huxley (HH) currents. We show that in all models, the instabilities of the asynchronous state can be classified in two classes. The first one consists of clustering instabilities, which exist in a restricted range of noise. These instabilities lead to synchronous patterns in which the population of neurons is broken into clusters of synchronously firing neurons. The irregularity of the firing patterns of the neurons is weak. The second class of instabilities, termed oscillatory firing rate instabilities, exists at any value of noise. They lead to cluster state at low noise. As the noise is increased, the instability occurs at larger coupling, and the pattern of firing that emerges becomes more irregular. In the regime of high noise and strong coupling, these instabilities lead to stochastic oscillations in which neurons fire in an approximately Poisson way with a common instantaneous probability of firing that oscillates in time.

Temporal processing in the exponential integrate-and-fire model is nonlinear

The exponential integrate-and-fire (EIF) model was introduced by Fourcaud-Trocmé et al. [How spike generation mechanisms determine the neuronal response to fluctuating inputs, J. Neurosci. 23 (2003) 11628–11640] as an extension of the standard leaky integrate-and-fire model. Here, the nonlinearity in the EIF model's temporal response to square-wave inputs is investigated. Comparing the time course of onset and offset responses revealed that offset responses have a steeper initial slope, but a slower approach to equilibrium. A linear systems analysis performed for square-wave inputs indicates that at frequencies above ∼40 Hz, gain was slightly smaller for square-wave inputs, but phase did not change significantly relative to simulations in which the corresponding sinusoids were presented in isolation.

Contributions of Intrinsic Membrane Dynamics to Fast Network Oscillations With Irregular Neuronal Discharges

During fast oscillations in the local field potential (40-100 Hz gamma, 100-200 Hz sharp-wave ripples) single cortical neurons typically fire irregularly at rates that are much lower than the oscillation frequency. Recent computational studies have provided a mathematical description of such fast oscillations, using the leaky integrate-and-fire (LIF) neuron model. Here, we extend this theoretical framework to populations of more realistic Hodgkin-Huxley-type conductance-based neurons. In a noisy network of GABAergic neurons that are connected randomly and sparsely by chemical synapses, coherent oscillations emerge with a frequency that depends sensitively on the single cell's membrane dynamics. The population frequency can be predicted analytically from the synaptic time constants and the preferred phase of discharge during the oscillatory cycle of a single cell subjected to noisy sinusoidal input. The latter depends significantly on the single cell's membrane properties and can be understood in the context of the simplified exponential integrate-and-fire (EIF) neuron. We find that 200-Hz oscillations can be generated, provided the effective input conductance of single cells is large, so that the single neuron's phase shift is sufficiently small. In a two-population network of excitatory pyramidal cells and inhibitory neurons, recurrent excitation can either decrease or increase the population rhythmic frequency, depending on whether in a neuron the excitatory synaptic current follows or precedes the inhibitory synaptic current in an oscillatory cycle. Detailed single-cell properties have a substantial impact on population oscillations, even though rhythmicity does not originate from pacemaker neurons and is an emergent network phenomenon.

Dynamics of the Instantaneous Firing Rate in Response to Changes in Input Statistics

We review and extend recent results on the instantaneous firing rate dynamics of simplified models of spiking neurons in response to noisy current inputs. It has been shown recently that the response of the instantaneous firing rate to small amplitude oscillations in the mean inputs depends in the large frequency limit f on the spike initiation dynamics. A particular simplified model, the exponential integrate-and-fire (EIF) model, has a response that decays as 1/f in the large frequency limit and describes very well the response of conductance-based models with a Hodgkin-Huxley type fast sodium current. Here, we show that the response of the EIF instantaneous firing rate also decays as 1/f in the case of an oscillation in the variance of the inputs for both white and colored noise. We then compute the initial transient response of the firing rate of the EIF model to a step change in its mean inputs and/or in the variance of its inputs. We show that in both cases the response speed is proportional to the neuron stationary firing rate and inversely proportional to a 'spike slope factor' Delta(T) that controls the sharpness of spike initiation: as 1/Delta(T) for a step change in mean inputs, and as 1/Delta(T) (2) for a step change in the variance in the inputs.