Starting from a spectral expansion of the Fokker-Plank equation for the membrane potential density in a network of spiking neurons, a low-dimensional dynamics of the collective firing rate is derived. As a result a $n$-order ordinary differential equation for the network activity can be worked out by taking into account the slowest $n$ modes of the expansion. The resulting low-dimensional dynamics naturally takes into account the strength of the synaptic couplings under the hypothesis of a not too fast changing membrane potential density. By considering only the two slowest modes, the firing rate dynamics is equivalent to the one of a damped oscillator in which the angular frequency and the relaxation time are state-dependent. The presented results apply to a wide class of networks of one-compartment neuron models.
We investigated the influence of efficacy of synaptic interaction on firing synchronization in excitatory neuronal networks. We found spike death phenomena, namely, the state of neurons transits from limit cycle to fixed point or transient state. The phenomena occur under the perturbation of excitatory synaptic interaction that has a high efficacy. We showed that the decrease of synaptic current results in spike death through depressing the feedback of sodium ionic current. In the networks with spike death property the degree of synchronization is lower and unsensitive to the heterogeneity of …
Fast-spiking (FS) interneurons in the brain are self-innervated by powerful inhibitory GABAergic autaptic connections. By computational modelling, we investigate how autaptic inhibition regulates the firing response of such interneurons. Our results indicate that autaptic inhibition both boosts the current threshold for action potential generation as well as modulates the input-output gain of FS interneurons. The autaptic transmission delay is identified as a key parameter that controls the firing patterns and determines multistability regions of FS interneurons. Furthermore, we observe that n…
A population of firing neurons is expected to carry information not only by mean firing rate but also by fluctuation and synchrony among neurons. In order to examine this possibility, we have studied responses of neuronal ensembles to three kinds of inputs: mean-, fluctuation- and synchrony-driven inputs. The generalized rate-code model including additive and multiplicative noise (H. Hasegawa, Phys. Rev. E {\bf 75} (2007) 051904) has been studied by direct simulations (DSs) and the augmented moment method (AMM) in which equations of motion for mean firing rate, fluctuation and synchrony are de…
The Abeles model of cortical activity assumes that in absence of stimulation neural activity in zero order can be described by a Poisson process. Here the model is extended to describe information processing by synfire chains within a network of activity uncorrelated to the synfire chain. A quantitative derivation of the transfer function from this concept is given.
Recurrent neural networks are powerful tools for understanding and modeling computation and representation by populations of neurons. Continuous-variable or "rate" model networks have been analyzed and applied extensively for these purposes. However, neurons fire action potentials, and the discrete nature of spiking is an important feature of neural circuit dynamics. Despite significant advances, training recurrently connected spiking neural networks remains a challenge. We present a procedure for training recurrently connected spiking networks to generate dynamical patterns autonomously, to p…
Neurons in the intact brain receive a continuous and irregular synaptic bombardment from excitatory and inhibitory pre-synaptic neurons, which determines the firing activity of the stimulated neuron. In order to investigate the influence of inhibitory stimulation on the firing time statistics, we consider Leaky Integrate-and-Fire neurons subject to inhibitory instantaneous post-synaptic potentials. In particular, we report exact results for the firing rate, the coefficient of variation and the spike train spectrum for various synaptic weight distributions. Our results are not limited to stimul…
Heterogeneity of neural attributes has recently gained a lot of attention and is increasing recognized as a crucial feature in neural processing. Despite its importance, this physiological feature has traditionally been neglected in theoretical studies of cortical neural networks. Thus, there is still a lot unknown about the consequences of cellular and circuit heterogeneity in spiking neural networks. In particular, combining network or synaptic heterogeneity and intrinsic heterogeneity has yet to be considered systematically despite the fact that both are known to exist and likely have signi…
The importance of self-feedback autaptic transmission in modulating spike-time irregularity is still poorly understood. By using a biophysical model that incorporates autaptic coupling, we here show that self-innervation of neurons participates in the modulation of irregular neuronal firing, primarily by regulating the occurrence frequency of burst firing. In particular, we find that both excitatory and electrical autapses increase the occurrence of burst firing, thus reducing neuronal firing regularity. In contrast, inhibitory autapses suppress burst firing and therefore tend to improve the r…
Maintaining the ability to fire sparsely is crucial for information encoding in neural networks. Additionally, spiking homeostasis is vital for spiking neural networks with changing numbers of weights and neurons. We discuss a range of network stabilisation approaches, inspired by homeostatic synaptic plasticity mechanisms reported in the brain. These include weight scaling, and weight change as a function of the network's spiking activity. We tested normalisation of the sum of weights for all neurons, and by neuron type. We examined how this approach affects firing rate and performance on clu…
We consider a wide class of spiking neuron models, defined by rather general set of conditions typical for basic models like leaky integrate and fire, or binding neuron model. A neuron is fed with a point renewal process. A relation between the three probability density functions (pdf): (i) pdf of input interspike intervals, (ii) pdf of output interspike intervals of a neuron with instantaneous feedback and (iii) pdf for that same neuron without feedback is derived. This allows to calculate any of the three pdfs provided the another two are given. Similar relation between corresponding means a…
We derive analytical formulae for the firing rate of integrate-and-fire neurons endowed with realistic synaptic dynamics. In particular we include the possibility of multiple synaptic inputs as well as the effect of an absolute refractory period into the description.
Strong inhibitory input to neurons, which occurs in balanced states of neural networks, increases synaptic current fluctuations. This has led to the assumption that inhibition contributes to the high spike-firing irregularity observed in vivo. We used single compartment neuronal models with time-correlated (due to synaptic filtering) and state-dependent (due to reversal potentials) input to demonstrate that inhibitory input acts to decrease membrane potential fluctuations, a result that cannot be achieved with simplified neural input models. To clarify the effects on spike-firing regularity, w…
In this work we consider a periodically forced generic integrate-and-fire model with a unique attracting equilibrium in the subthreshold dynamics and study the dependence of the firing-rate on the frequency of the drive. In an earlier study we have obtained rigorous results on the bifurcation structure in such systems, with emphasis on the relation between the firing-rate and the rotation number of the existing periodic orbits. In this work we study how these bifurcation structures behave upon variation of the frequency of the input. This allows us to show that the dependence of the firing-rat…
A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the…
In this paper, we examine the effects of correlated Gaussian noise on a two-dimensional neuronal network that is locally modeled by the Rulkov map. More precisely, we study the effects of the noise correlation on the variations of the mean firing rate and the correlations among neurons versus the noise intensity. Via numerical simulations, we show that the mean firing rate can always be optimized at an intermediate noise intensity, irrespective of the noise correlation. On the other hand, variations of the population coherence with respect to the noise intensity are strongly influenced by the …
The background activity of a cortical neural network is modeled by a homogeneous integrate-and-fire network with unreliable inhibitory synapses. Numerical and analytical calculations show that the network relaxes into a stationary state of high attention. The majority of the neurons has a membrane potential just below the threshold; as a consequence the network can react immediately - on the time scale of synaptic transmission- on external pulses. The neurons fire with a low rate and with a broad distribution of interspike intervals. Firing events of the total network are correlated over short…
We compute the firing rate of a leaky integrate-and-fire (LIF) neuron with stochastic conductance-based inputs in the limit when synaptic decay times are much shorter than the membrane time constant. A comparison of our analytical results to numeric simulations is presented for a range of biophysically-realistic parameters.
Periodic and nonperiodic components of electrophysiological signals are modelled in terms of syncronized sequences of closed loops of firing neurons correlated in Markov chains. Single closed loops of firing neurons reproduce fundamental and harmonic components, appearing as lines in the power spectra at frequencies ranging about from $0.5 Hz$ to $ 100 Hz$. Further interesting features of the brainwave signals emerge by considering multiple syncronized sequences of closed loops. In particular, we show that the fluctuations of the number of syncronized loops leads to the onset of broadband powe…
A central question in neuroscience is to understand how noisy firing patterns are used to transmit information. Because neural spiking is noisy, spiking patterns are often quantified via pairwise correlations, or the probability that two cells will spike coincidentally, above and beyond their baseline firing rate. One observation frequently made in experiments, is that correlations can increase systematically with firing rate. Theoretical studies have determined that stimulus-dependent correlations that increase with firing rate can have beneficial effects on information coding; however, we st…
The dominant modeling framework for understanding cortical computations are heuristic firing rate models. Despite their success, these models fall short to capture spike synchronization effects, to link to biophysical parameters and to describe finite-size fluctuations. In this opinion article, we propose that the refractory density method (RDM), also known as age-structured population dynamics or quasi-renewal theory, yields a powerful theoretical framework to build rate-based models for mesoscopic neural populations from realistic neuron dynamics at the microscopic level. We review recent ad…
Spiking neural networks (SNNs) has attracted much attention due to its great potential of modeling time-dependent signals. The firing rate of spiking neurons is decided by control rate which is fixed manually in advance, and thus, whether the firing rate is adequate for modeling actual time series relies on fortune. Though it is demanded to have an adaptive control rate, it is a non-trivial task because the control rate and the connection weights learned during the training process are usually entangled. In this paper, we show that the firing rate is related to the eigenvalue of the spike gene…
Effects of distractions such as noises and parameter heterogeneity have been studied on the firing activity of ensemble neurons, each of which is described by the extended Morris-Lecar model showing the graded persisting firings with the aid of an included ${\rm Ca}^{2+}$-dependent cation current. Although the sustained activity of {\it single} neurons is rather robust in a sense that the activity is realized even in the presence of the distractions, the graded frequency of sustained firings is vulnerable to them. It has been shown, however, that the graded persisting activity of {\it ensemble…
Mounting evidence in recent years suggests that astrocytes, a sub-type of glial cells, not only serve metabolic and structural support for neurons and synapses but also play critical roles in regulation of proper functioning of the nervous system. In this work, we investigate the effect of astrocyte on the spontaneous firing activity of a neuron through a combined model which includes a neuron-astrocyte pair. First, we show that an astrocyte may provide a kind of multistability in neuron dynamics by inducing different firing modes such as random and bursty spiking. Then, we identify the underl…
We examine the effects of stochastic input currents on the firing behavior of two excitable neurons coupled with fast excitatory synapses. In such cells (models), typified by the quadratic integrate and fire model, mutual synaptic coupling can cause sustained firing or oscillatory behavior which is necessarily antiphase. Additive Gaussian white noise can transiently terminate the oscillations, hence destroying the stable limit cycle. Further application of the noise may return the system to spiking activity. In a particular noise range, the transition times between the oscillating and the rest…
On the basis of the evident ability of neuronal olfactory systems to evaluate the intensity of an odorous stimulus and at the same time also recognise the identity of the odorant over a large range of concentrations, a few biologically-realistic hypotheses on some of the underlying neural processes are made. In particular, it is assumed that the receptor neurons mean firing-rate scale monotonically with odorant intensity, and that the receptor sensitivities range widely across odorants and receptor neurons hence leading to highly distributed representations of the stimuli. The mathematical imp…
Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models ---also known as firing rate models, or firing rate equations--- to account for electrical synapses. Here we introduce a novel firing rate model that exactly describes the mean field dynamics of heterogeneous populations of quadratic integra…
Autapses are synapses that connect a neuron to itself in the nervous system. Previously, both experimental and theoretical studies have demonstrated that autaptic connections in the nervous system have a significant physiological function. Autapses in nature provide self-delayed feedback, thus introducing an additional time scale to neuronal activities and causing many dynamic behaviors in neurons. Recently, theoretical studies have revealed that an autapse provides a control option for adjusting the response of a neuron: e.g., an autaptic connection can cause the electrical activities of the …
Even a single neuron may be able to produce significant lognormal features in its firing statistics due to noise in the charging ion current. A mathematical scheme introduced in advanced nanotechnology is relevant for the analysis of this mechanism in the simplest case, the integrate-and-fire model with white noise in the charging ion current.
When attention is directed into the receptive field of a V4 neuron, its contrast response curve is shifted to lower contrast values (Reynolds et al, 2000, Neuron 26:703). Attention also increases the coherence between neurons responding to the same stimulus (Fries et al, 2001, Science 291:1560). We studied how the firing rate and synchrony of a densely interconnected cortical network varied with contrast and how they were modulated by attention. We found that an increased driving current to the excitatory neurons increased the overall firing rate of the network, whereas variation of the drivin…
Correlation-based Hebbian plasticity is thought to shape neuronal connectivity during development and learning, whereas homeostatic plasticity would stabilize network activity. Here we investigate another, new aspect of this dichotomy: Can Hebbian associative properties also emerge as a network effect from a plasticity rule based on homeostatic principles on the neuronal level? To address this question, we simulated a recurrent network of leaky integrate-and-fire neurons, in which excitatory connections are subject to a structural plasticity rule based on firing rate homeostasis. We show that …
The state space of a conventional Hopfield network typically exhibits many different attractors of which only a small subset satisfy constraints between neurons in a globally optimal fashion. It has recently been demonstrated that combining Hebbian learning with occasional alterations of normal neural states avoids this problem by means of self-organized enlargement of the best basins of attraction. However, so far it is not clear to what extent this process of self-optimization is also operative in real brains. Here we demonstrate that it can be transferred to more biologically plausible neur…
Neuronal networks are controlled by a combination of the dynamics of individual neurons and the connectivity of the network that links them together. We study a minimal model of the preBotzinger complex, a small neuronal network that controls the breathing rhythm of mammals through periodic firing bursts. We show that the properties of a such a randomly connected network of identical excitatory neurons are fundamentally different from those of uniformly connected neuronal networks as described by mean-field theory. We show that (i) the connectivity properties of the networks determines the loc…
Neuronal models based on the Hodgkin-Huxley equation form a fundamental framework in the field of computational neuroscience. While the neuronal state is often modeled deterministically, experimental recordings show stochastic fluctuations, presumably driven by molecular noise from the underlying microphysical conditions. In turn, the firing of individual neurons gives rise to an electric field n extracellular space, also thought to affect the firing pattern of nearby neurons. We develop a multiscale model which combines a stochastic ion channel gating process taking place on the neuronal memb…
Spike correlations between neurons are ubiquitous in the cortex, but their role is at present not understood. Here we describe the firing response of a leaky integrate-and-fire neuron (LIF) when it receives a temporarily correlated input generated by presynaptic correlated neuronal populations. Input correlations are characterized in terms of the firing rates, Fano factors, correlation coefficients and correlation timescale of the neurons driving the target neuron. We show that the sum of the presynaptic spike trains cannot be well described by a Poisson process. Solutions of the output firing…
In this correspondence information theoretical tools are used to investigate the statistical properties of modeled cochlear nucleus globular bushy cell spike trains. The firing patterns are obtained from a simulation software that generates sample spike trains from any auditory input. Here we analyze for the first time the responses of globular bushy cells to voiced and unvoiced speech sounds. Classical entropy estimates, such as the direct method, are improved upon by considering a time-varying and time-dependent entropy estimate. With this method we investigated the relationship between the …
Definition. The phase-of-firing code is a neural coding scheme whereby neurons encode information using the time at which they fire spikes within a cycle of the ongoing oscillatory pattern of network activity. This coding scheme may allow neurons to use their temporal pattern of spikes to encode information that is not encoded in their firing rate.
We investigate numerically the collective dynamical behavior of pulse-coupled non-leaky integrate-and-fire-neurons that are arranged on a two-dimensional small-world network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Beside a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulent-like patterns, which -- depending on netw…
We studied the impact of a dynamical threshold on the f-I curve-the relationship between the input and the firing rate of a neuron-in the presence of background synaptic inputs. First, we found that, while the leaky integrate-and-fire model cannot reproduce the f-I curve of a cortical neuron, the leaky integrate-and-fire model with dynamical threshold can reproduce it very well. Second, we found that the dynamical threshold modulates the onset and the asymptotic behavior of the f-I curve. These results suggest that a cortical neuron has an adaptation mechanism and that the dynamical threshold …
The coding properties of cells with different types of receptive fields have been studied for decades. ON-type neurons fire in response to positive fluctuations of the time-dependent stimulus, whereas OFF cells are driven by negative stimulus segments. Biphasic cells, in turn, are selective to up/down or down/up stimulus upstrokes. In this paper, we explore the way in which different receptive fields affect the firing statistics of Poisson neuron models, when driven with slow stimuli. We find analytical expressions for the time-dependent peri-stimulus time histogram and the inter-spike interva…
Two recent experimental observations pose a challenge to many cortical models. First, the activity in the auditory cortex is sparse, and firing rates can be described by a lognormal distribution. Second, the distribution of non-zero synaptic strengths between nearby cortical neurons can also be described by a lognormal distribution. Here we use a simple model of cortical activity to reconcile these observations. The model makes the experimentally testable prediction that synaptic efficacies onto a given cortical neuron are statistically correlated, i.e. it predicts that some neurons receive ma…
Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixpoint and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increa…
Realizations of low firing rates in neural networks usually require globally balanced distributions among excitatory and inhibitory links, while feasibility of temporal coding is limited by neuronal millisecond precision. We show that cooperation, governing global network features, emerges through nodal properties, as opposed to link distributions. Using in vitro and in vivo experiments we demonstrate microsecond precision of neuronal response timings under low stimulation frequencies, whereas moderate frequencies result in a chaotic neuronal phase characterized by degraded precision. Above a …
The effect of environmental temperature on neuronal spiking behaviors is investigated by numerically simulating the temperature dependence of spiking threshold of the Hodgkin-Huxley neuron subject to synaptic stimulus. We find that the spiking threshold exhibits a global minimum in a "comfortable temperature" range where spike initiation needs weakest synaptic strength, indicating the occurrence of optimal use of synaptic transmission in neural system. We further explore the biophysical origin of this phenomenon in ion channel gating kinetics and also discuss its possible biological relevance …
Spiking neural networks (SNNs) communicate through the all-or-none spiking activity of neurons. However, fitting the large number of SNN model parameters to observed neural activity patterns, for example, in biological experiments, remains a challenge. Previous work using genetic algorithm (GA) optimisation on a specific efficient SNN model, using the Izhikevich neuronal model, was limited to a single parameter and objective. This work applied a version of GA, called non-dominated sorting GA (NSGA-III), to demonstrate the feasibility of performing multi-objective optimisation on the same SNN, …
Repeated occurrences of serial firing sequences of a group of neurons with fixed time delays between neurons are observed in many experiments involving simultaneous recordings from multiple neurons. Such temporal patterns are potentially indicative of underlying microcircuits and it is important to know when a repeatedly occurring pattern is statistically significant. These sequences are typically identified through correlation counts, such as in the two-tape algorithm of Abeles and Gerstein. In this paper we present a method for deciding on the significance of such correlations by characteriz…
Networks in the brain consist of different types of neurons. Here we investigate the influence of neuron diversity on the dynamics, phase space structure and computational capabilities of spiking neural networks. We find that already a single neuron of a different type can qualitatively change the network dynamics and that mixed networks may combine the computational capabilities of ones with a single neuron type. We study inhibitory networks of concave leaky (LIF) and convex "anti-leaky" (XIF) integrate-and-fire neurons that generalize irregularly spiking non-chaotic LIF neuron networks. Endo…
In this paper, we clarify the mechanisms underlying a general phenomenon present in pulse-coupled heterogeneous inhibitory networks: inhibition can induce not only suppression of the neural activity, as expected, but it can also promote neural reactivation. In particular, for globally coupled systems, the number of firing neurons monotonically reduces upon increasing the strength of inhibition (neurons' death). However, the random pruning of the connections is able to reverse the action of inhibition, i.e. in a sparse network a sufficiently strong synaptic strength can surprisingly promote, ra…
We present a simple Markov model of spiking neural dynamics that can be analytically solved to characterize the stochastic dynamics of a finite-size spiking neural network. We give closed-form estimates for the equilibrium distribution, mean rate, variance and autocorrelation function of the network activity. The model is applicable to any network where the probability of firing of a neuron in the network only depends on the number of neurons that fired in a previous temporal epoch. Networks with statistically homogeneous connectivity and membrane and synaptic time constants that are not exces…
We propose a computational model of neuron, called firing cell (FC), properties of which cover such phenomena as attenuation of receptors for external stimuli, delay and decay of postsynaptic potentials, modification of internal weights due to propagation of postsynaptic potentials through the dendrite, modification of properties of the analog memory for each input due to a pattern of short-time synaptic potentiation or long-time synaptic potentiation (LTP), output-spike generation when the sum of all inputs exceeds a threshold, and refraction. The cell may take one of the three forms: excitat…
The impact of inhibitory and excitatory synapses in delay-coupled Hodgkin--Huxley neurons that are driven by noise is studied. If both synaptic types are used for coupling, appropriately tuned delays in the inhibition feedback induce multiple firing coherence resonances at sufficiently strong coupling strengths, thus giving rise to tongues of coherency in the corresponding delay-strength parameter plane. If only inhibitory synapses are used, however, appropriately tuned delays also give rise to multiresonant responses, yet the successive delays warranting an optimal coherence of excitations ob…
