Recently a method has been put forward to connect the measures of spontaneous neuronal activity and the measures of the average single-neuron response to stimuli via fluctuation-response relations (FRRs) for some integrate-and-fire (IF) type neuron models. In this work we expand this method to populations of neurons, relating their spontaneous correlation and linear-response statistics. To this end, we analyze the simple case of uncoupled cells modeled by IF neurons (first stage of processing) which receive common stochastic input and project their output spike trains onto a readout neuron (second stage of processing). We derive and verify FRRs connecting the single neuron response to cross-correlations among neurons and the response of the full system to cross-stage correlations. Furthermore, we utilize these FRRs to derive approximations of all cross-stage cross-spectra for a relevant model of a second-stage cell, the partial synchronous output (PSO). We conclude with a discussion of how our results can be expanded to more involved network settings and neuron models.

Abstract Recurrent spiking neural networks (RSNNs) can be implemented very efficiently in neuromorphic systems. Nevertheless, training of these models with powerful gradient-based learning algorithms is mostly performed on standard digital hardware using Backpropagation through time (BPTT). However, BPTT has substantial limitations. It does not permit online training and its memory consumption scales linearly with the number of computation steps. In contrast, learning methods using forward propagation of gradients operate in an online manner with a memory consumption independent of the number of time steps. These methods enable SNNs to learn from continuous, infinite-length input sequences. In addition, approximate forward propagation algorithms have been developed that can be implemented on neuromorphic hardware. Yet, slow execution speed on conventional hardware as well as inferior performance has hindered their widespread application. In this work, we introduce HYbrid PRopagation (HYPR) that combines the efficiency of parallelization with approximate online forward learning. Our algorithm yields high-throughput online learning through parallelization, paired with constant, i.e., sequence length independent, memory demands. HYPR enables parallelization of parameter update computation over subsequences for RSNNs consisting of almost arbitrary non-linear spiking neuron models. We apply HYPR to networks of spiking neurons with oscillatory subthreshold dynamics. We find that this type of neuron model is particularly well trainable by HYPR, resulting in an unprecedentedly low task performance gap between approximate forward gradient learning and BPTT.

Abstract The regulation of balanced dopaminergic neural activities between direct and indirect pathways in the putamen of the striatum plays a crucial role in maintaining daily healthy movements. An imbalanced dopaminergic neural activity in the putamen is known to cause neurological movement disorders such as Parkinson’s disease (PD) due to dopamine depletion. Deep brain stimulation (DBS) is a strong candidate for clinical treatments for drug-resistant PD. However, little is known about how the parameters of electric stimuli can be set systematically to alleviate PD symptoms. We tested the hypothesis that an optimal periodic electric stimulus can be found to compensate for the imbalanced spiking activities between direct and indirect pathways caused by weak subthreshold dopaminergic input from substantia nigra pars compacta (SNc). This stimulation is based on the concept of vibrational resonance (VR) or pulse-frequency-dependent resonance (PFDR), and studied through computer simulations. Sinusoidal or pulsatile electric stimulus was delivered to a pathway from SNc in a striatal neural network model with a population of Izhikevich type neuron models, and a balance index of the spike firing rates in direct and indirect pathways was calculated as a function of sinusoidal or pulse frequency. The results of computer simulations show that an optimal stimulation frequency to balance spiking activities between direct and indirect pathways in the putamen is observed for VR and PFDR. These findings could contribute to the determination of an optimal strategy to find the stimulating parameters for closed-loop DBS control in PD.

LDN-SNP: SNP-based lightweight deep network for CT image segmentation of COVID-19

Two types of neuron models are constructed in this paper, namely the single discrete memristive synaptic neuron model and the dual discrete memristive synaptic neuron model. Firstly, it is proved that both models have only one unstable equilibrium point. Then, the influence of the coupling strength parameters and neural membrane amplification coefficient of the corresponding system of the two models on the rich dynamical behavior of the systems is analyzed. Research has shown that when the number of discrete local active memristor used as simulation synapses in the system increases from one to two, the coupling strength parameter of the same memristor has significantly different effects on the dynamical behavior of the system within the same range, that is, from a state with periodicity, chaos, and periodicity window to a state with only chaos. In addition, under the influence of coupling strength parameters and neural membrane amplification coefficients, the complexity of the system weakens to varying degrees. Moreover, under the effect of two memristors, the system exhibits a rare and interesting phenomenon where the coupling strength parameter and the neural membrane amplification coefficient can mutually serve as control parameter, resulting in the generation of a remerging Feigenbaum tree. Finally, the pseudo-randomness of the chaotic systems corresponding to the two models are detected by NIST SP800-22, and relevant simulation results are verified on the DSP hardware experimental platform. The discrete memristive synaptic neuron models established in this article provide assistance in studying the relevant working principles of real neurons.

Though there are many neuron models based on differential equations, the complexity in realizing them into digital circuits is still a challenge. Hence, many new discrete neuron models have been recently proposed, which can be easily implemented in digital circuits. We consider the well-known FitzHugh-Nagumo model and derive the discrete version of the model considering the sigmoid type of recovery variable and electromagnetic flux coupling. We show the various time series plots confirming the existence of periodic and chaotic bursting as in differential equation type neuron models. Also, we have used the bifurcation plots, Lyapunov exponents, and frequency bifurcations to investigate the dynamics of the proposed discrete neuron model. Different topologies of networks like single, two, and three layers are considered to analyze the wave propagation phenomenon in the network. We introduce the concept of using energy levels of nodes to study the spiral wave existence and compare them with the spatiotemporal snapshots. Interestingly, the energy plots clearly show that when the energy level of nodes is different and distributed, the occurrence of the spiral waves is identified in the network.

A fundamental challenge for the theoretical study of neuronal networks is to make the link between complex biophysical models based directly on experimental data, to progressively simpler mathematical models that allow the derivation of general operating principles. We present a strategy that successively maps a relatively detailed biophysical population model, comprising conductance-based Hodgkin-Huxley type neuron models with connectivity rules derived from anatomical data, to various representations with fewer parameters, finishing with a firing rate network model that permits analysis. We apply this methodology to primary visual cortex of higher mammals, focusing on the functional property of stimulus orientation selectivity of receptive fields of individual neurons. The mapping produces compact expressions for the parameters of the abstract model that clearly identify the impact of specific electrophysiological and anatomical parameters on the analytical results, in particular as manifested by specific functional signatures of visual cortex, including input-output sharpening, conductance invariance, virtual rotation and the tilt after effect. Importantly, qualitative differences between model behaviours point out consequences of various simplifications. The strategy may be applied to other neuronal systems with appropriate modifications.

Network performance of neurons plays a vital role in determining the behavior of many physiological systems. In this paper, we discuss the wave propagation phenomenon in a network of neurons considering obstacles in the network. Numerous studies have shown the disastrous effects caused by the heterogeneity induced by the obstacles, but these studies have been mainly discussing the orientation effects. Hence, we are interested in investigating the effects of both the size and orientation of the obstacles in the wave re-entry and spiral wave formation in the network. For this analysis, we have considered two types of neuron models and a pancreatic beta cell model. In the first neuron model, we use the well-known differential equation-based neuron models, and in the second type, we used the hybrid neuron models with the resetting phenomenon. We have shown that the size of the obstacle decides the spiral wave formation in the network and horizontally placed obstacles will have a lesser impact on the wave re-entry than the vertically placed obstacles.

Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < α ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.

Random pulses contribute to stochastic resonance in neuron models, whereas common random pulses cause stochastic-synchronized excitation in uncoupled neuron models. We studied concurrent phenomena contributing to phase synchronization and stochastic resonance following induction by a weak common random pulse in uncoupled non-identical Hodgkin-Huxley type neuron models. The common random pulse was selected from a gamma distribution and the degree of synchronization depended on the corresponding shape parameter. Specifically, a low shape parameter of the weak random pulse induced well-synchronized spiking in uncoupled neuron models, whereas a high shape parameter of the weak random pulse or a weak periodic pulse caused low degrees of synchronization. These were improved by concurrent inputs of periodic and random pulses with high shape parameters. Finally, the output pulse was synchronized with the periodic pulse, and the common random pulse revealed periodic responses in the present neuron models.

A piece-wise constant (PWC) neuron model is an electronic circuit neuron model, which has a PWC vector field. In this brief, a novel generalized PWC neuron model is presented in order to reproduce fundamental bifurcation mechanisms of bursting observed in a Hodgkin–Huxley type neuron model. Thanks to the PWC vector field, the bifurcation mechanisms of the presented model can be analyzed theoretically. It is then shown that the theoretical analysis results are useful for efficient design of the presented model. A so designed model is implemented and circuit experiments validate occurrences of the bursting and the bifurcation.

The aim of this work is to investigate the role of network organization and the neuron model type on the collective dynamic behavior of striatal population. For that purpose, two different scale neuron models which are phenomenological Izhikevich and conductance-based Hodgkin-Huxley (HH) type are used to investigate the dynamic behavior of MS neurons. Two network architectures are proposed with inhibitory and excitatory synaptic currents. In these networks, while all MS neurons affect each other with inhibitory synaptic currents, an excitatory current is applied to all MS neurons in the first layer, to represent the cortical inputs. A mathematical model of a medium spiny neuron of striatum based on HH type neuron model is proposed using different calcium channels and its dynamical behavior is investigated. It is observed that when the original HH model is used, regular spiking type behavior is observed. Including the high threshold calcium current, after hyperpolarization calcium current and voltage gated potasium current into the model improves the modeling capabilities. With extended ion channels, in addition to regular spiking behavior, bursting with resting stage are obtained. Then, Izhikevich neuron model is used in the network structures to compare the dynamic behaviors and computational time.

Improving the quality of extracting dynamics from interspike intervals via a resampling approach

In vitro studies have previously found a class of vestibular nuclei neurons to exhibit a bidirectional afterhyperpolarization (AHP) in their membrane potential, due to calcium and calcium-activated potassium conductances. More recently in vivo studies of such vestibular neurons were found to exhibit a boosting nonlinearity in their input-output tuning curves. In this paper, a Hodgkin-Huxley (HH) type neuron model, originally developed to reproduce the in vitro AHP, is shown to produce a boosting nonlinearity similar to that seen in vivo for increased the calcium conductance. Indicative of a bifurcation, the HH model is reduced to a generalized integrate-and-fire (IF) model that preserves the bifurcation structure and boosting nonliearity. By then projecting the neuron model’s phase space trajectories into 2D, the underlying geometric mechanism relating the AHP and boosting nonlinearity is revealed. Further simplifications and approximations are made to derive analytic expressions for the steady steady state firing rate as a function of bias current, μ, as well as the gain (i.e. its slope) and the position of its peak at μ = μ*. Finally, although the boosting nonlinearity has not yet been experimentally observed in vitro, testable predictions indicate how it might be found.

Many circuits of the brain can be described by a system of interacting neural populations that are approximately homogeneous. For instance, cortical layers typically consist of a few main types of excitatory and inhibitory neurons that form small homogeneous populations of neurons. Such systems can be modeled on different spatial scales. On the microscopic scale, single cell activity has been faithfully described by reduced phenomenological neuron models [1]. Simulations of networks of such neuron models are, however, computationally expensive and do not offer much analytical insight. On the other hand, mesoscopic population models are reduced descriptions of the global activities of each population. These activities are stochastic due to the finite sizes of the populations. Mesoscopic models can be efficiently simulated and provide a better understanding of the dynamics owing to the abstraction of microscopic information. However, it is largely unknown how to relate mesoscopic population models to microscopic properties such as neural refractoriness, synaptic conductance dynamics and spike-frequency adaptation. Here, we derive a mesoscopic population model for microscopic networks of generalized integrate-and-fire neurons [1]. This type of neuron model supports important properties like neural refractoriness, multiple-time-scale adaptation, stochastic spike generation and synaptic dynamics; its parameters can be directly extracted from experiments of real neurons. In particular, we use a mean-field and a quasi-renewal approximation [1] to derive stochastic integral equations for the population rates. These equations highlight how the history of activities and fluctuations affects the refractoriness of the populations and the activities at the current time. They can be solved forward in time and thus allow to quickly generate stochastic samples of spontaneous or evoked activities (Fig. 1B,C) that have the same statistics as a corresponding microscopic network simulation to a high degree of accuracy (Fig. ​(Fig.1D).1D). The theory not only captures linear population dynamics [2] but also nonlinear collective effects that emerge on the population level such as metastability (Fig. ​(Fig.1).1). Our novel theory establishes a general framework for modeling neural population dynamics based on microscopic neuronal parameters. It offers an efficient way to analyze cortical circuits and its computational functions, and how they depend on single-cell and synaptic properties. Figure 1 Stochastic population equation precisely captures the collective bistable dynamics of a spiking neural network. A Two mutually inhibitory populations of 500 neurons each. B,C Sample paths of the spiking neural network and the derived population model, ...

Studying ion channel currents generated distally from the recording site is difficult because of artifacts caused by poor space clamp and membrane filtering. A computational model can quantify artifact parameters for correction by simulating the currents only if their exact anatomical location is known. We propose that the same artifacts that confound current recordings can help pinpoint the source of those currents by providing a signature of the neuron’s morphology. This method can improve the recording quality of currents initiated at the spike initiation zone (SIZ) that are often distal to the soma in invertebrate neurons. Drosophila being a valuable tool for characterizing ion currents, we estimated the SIZ location and quantified artifacts in an identified motoneuron, aCC/MN1-Ib, by constructing a novel multicompartmental model. Initial simulation of the measured biophysical channel properties in an isopotential Hodgkin-Huxley type neuron model partially replicated firing characteristics. Adding a second distal compartment, which contained spike-generating Na+ and K+ currents, was sufficient to simulate aCC’s in vivo activity signature. Matching this signature using a reconstructed morphology predicted that the SIZ is on aCC’s primary axon, 70 μm after the most distal dendritic branching point. From SIZ to soma, we observed and quantified selective morphological filtering of fast activating currents. Non-inactivating K+ currents are filtered ∼3 times less and despite their large magnitude at the soma they could be as distal as Na+ currents. The peak of transient component (NaT) of the voltage-activated Na+ current is also filtered more than the magnitude of slower persistent component (NaP), which can contribute to seizures. The corrected NaP/NaT ratio explains the previously observed discrepancy when the same channel is expressed in different cells. In summary, we used an in vivo signature to estimate ion channel location and recording artifacts, which can be applied to other neurons.

While correlated activity is observed ubiquitously in the brain, its role in neural coding has remained controversial. Recent experimental results have demonstrated that correlated but not single-neuron activity can encode the detailed time course of the instantaneous amplitude (i.e., envelope) of a stimulus. These have furthermore demonstrated that such coding required and was optimal for a nonzero level of neural variability. However, a theoretical understanding of these results is still lacking. Here we provide a comprehensive theoretical framework explaining these experimental findings. Specifically, we use linear response theory to derive an expression relating the correlation coefficient to the instantaneous stimulus amplitude, which takes into account key single-neuron properties such as firing rate and variability as quantified by the coefficient of variation. The theoretical prediction was in excellent agreement with numerical simulations of various integrate-and-fire type neuron models for various parameter values. Further, we demonstrate a form of stochastic resonance as optimal coding of stimulus variance by correlated activity occurs for a nonzero value of noise intensity. Thus, our results provide a theoretical explanation of the phenomenon by which correlated but not single-neuron activity can code for stimulus amplitude and how key single-neuron properties such as firing rate and variability influence such coding. Correlation coding by correlated but not single-neuron activity is thus predicted to be a ubiquitous feature of sensory processing for neurons responding to weak input.

Optimization techniques have been applied to neuron models for a variety of purposes, including control of neuron firing rates and minimizing input stimulus current magnitudes. Optimal control is used to minimize a quantity of interest; often, the time or energy needed to complete an objective. Rather than attempting to control or modify neuron dynamics, this paper demonstrates that optimal control can be used to obtain an optimal input stimulus current i*(t) which causes a six dimensional Hodgkin–Huxley type neuron model to approximate a specified reference membrane voltage. The reference voltages considered in this paper consist of one or more action potentials as evoked by an input current i(t). In the described method, the user prescribes a balance of low squared integral of input stimulus current (input stimulus “energy”) and accurate tracking of the original reference voltage. In a previous work, the authors applied this approach to a reduced order neuron model. This paper demonstrates the applicability of this technique to biologically plausible higher dimensional conductance based neuron models. For each investigated neuron response, the method discovered optimal input stimuli current i*(t) having a lower energy than the original i(t), while still providing accurate tracking of the reference voltage.

Global parameter estimation of an Hodgkin–Huxley formalism using membrane voltage recordings: Application to neuro-mimetic analog integrated circuits

Exposed to a sufficiently high extracellular potassium concentration ([K( + )]₀), the neuron can fire spontaneous discharges or even become inactivated due to membrane depolarisation ('depolarisation block'). Since these phenomena likely are related to the maintenance and propagation of seizure discharges, it is of considerable importance to understand the conditions under which excess [K( + )]₀ causes them. To address the putative effect of glial buffering on neuronal activity under elevated [K( + )](o) conditions, we combined a recently developed dynamical model of glial membrane ion and water transport with a Hodgkin-Huxley type neuron model. In this interconnected glia-neuron model we investigated the effects of natural heterogeneity or pathological changes in glial membrane transporter density by considering a large set of models with different, yet empirically plausible, sets of model parameters. We observed both the high [K( + )]₀-induced duration of spontaneous neuronal firing and the prevalence of depolarisation block to increase when reducing the magnitudes of the glial transport mechanisms. Further, in some parameter regions an oscillatory bursting spiking pattern due to the dynamical coupling of neurons and glia was observed. Bifurcation analyses of the neuron model and of a simplified version of the neuron-glia model revealed further insights about the underlying mechanism behind these phenomena. The above insights emphasise the importance of combining neuron models with detailed astroglial models when addressing phenomena suspected to be influenced by the astroglia-neuron interaction. To facilitate the use of our neuron-glia model, a CellML version of it is made publicly available.