Single Neuron Model Research Articles

HOPF BIFURCATION AND CHAOS IN TABU LEARNING NEURON MODELS

In this paper, we consider the nonlinear dynamical behaviors of some tabu learning neuron models. We first consider a tabu learning single neuron model. By choosing the memory decay rate as a bifurcation parameter, we prove that Hopf bifurcation occurs in the neuron. The stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory. We give a numerical example to verify the theoretical analysis. Then, we demonstrate the chaotic behavior in such a neuron with sinusoidal external input, via computer simulations. Finally, we study the chaotic behaviors in tabu learning two-neuron models, with linear and quadratic proximity functions respectively.

Synchronization and Spindle Oscillation in Noisy Integrate-and-Fire-or-Burst Neurons with Inhibitory Coupling

We propose another integrate-and-fire model as a single neuron model. We study a globally coupled noisy integrate-and-fire model with inhibitory interaction using the Fokker-Planck equation and the Langevin equation, and find a reentrant transition of oscillatory states. Intermittent time evolutions of neuron firing are found in strongly inhibited systems. We propose another integrate-and-fire-or-burst model including the dynamics of the low-threshold Ca$^{2+}$ current based on the new integrate-and-fire model. We study a globally coupled noisy integrate-and-fire-or-burst model with inhibitory interaction using the Fokker-Planck equation, and find bistability of the tonic mode and burst mode. Doubly periodic oscillation appears in a coupled system of two neuron assemblies, which is similar to the spindle oscillation in thalamic cells.

History-dependent multiple-time-scale dynamics in a single-neuron model.

History-dependent characteristic time scales in dynamics have been observed at several levels of organization in neural systems. Such dynamics can provide powerful means for computation and memory. At the level of the single neuron, several microscopic mechanisms, including ion channel kinetics, can support multiple-time-scale dynamics. How the temporally complex channel kinetics gives rise to dynamical properties of the neuron is not well understood. Here, we construct a model that captures some features of the connection between these two levels of organization. The model neuron exhibits history-dependent multiple-time-scale dynamics in several effects: first, after stimulation, the recovery time scale is related to the stimulation duration by a power-law scaling; second, temporal patterns of neural activity in response to ongoing stimulation are modulated over time; finally, the characteristic time scale for adaptation after a step change in stimulus depends on the duration of the preceding stimulus. All these effects have been observed experimentally and are not explained by current single-neuron models. The model neuron here presented is composed of an ensemble of ion channels that can wander in a large pool of degenerate inactive states and thus exhibits multiple-time-scale dynamics at the molecular level. Channel inactivation rate depends on recent neural activity, which in turn depends through modulations of the neural response function on the fraction of active channels. This construction produces a model that robustly exhibits nonexponential history-dependent dynamics, in qualitative agreement with experimental results.

A single compartment neuron model with activity-dependent conductances during NMDA induced activity

In most neuron models the values of maximal conductances of membrane ionic currents are fixed. In our paper we investigate spiking activity of the neuron model activated tonically by NMDA synapse, when the membrane ionic currents are dynamically dependent on calcium concentration, as in a model by Abbott and coauthors (1993). A spiking neuron model (in Matlab/Simulink environment) is based on the properties of lamprey spinal neurons. The basic neuron is a one-compartment model with voltage-gated Na+, K+, Ca2+, KCa+ channels. The Na+ and K+ currents are described with the dynamic equations of Hodgkin-Huxley model (Hodgkin and Huxley 1952). The Ca2+ and KCa+ channels are modeled using description of calcium dynamic introduced by Ekeberg and coauthors (1991). The model was tonically activated by NMDA synapse described by a kinetic model of synaptic transmission. We analyzed the activity of this model and showed that when only one of conductances is calcium-dependent, the cell is not able to react to and recover from external perturbations.

Delayed Inhibition in Cortical Receptive Fields and the Discrimination of Complex Stimuli

Although auditory cortex is thought to play an important role in processing complex natural sounds such as speech and animal vocalizations, the specific functional roles of cortical receptive fields (RFs) remain unclear. Here, we study the relationship between a behaviorally important function: the discrimination of natural sounds and the structure of cortical RFs. We examine this problem in the model system of songbirds, using a computational approach. First, we constructed model neurons based on the spectral temporal RF (STRF), a widely used description of auditory cortical RFs. We focused on delayed inhibitory STRFs, a class of STRFs experimentally observed in primary auditory cortex (ACx) and its analog in songbirds (field L), which consist of an excitatory subregion and a delayed inhibitory subregion cotuned to a characteristic frequency. We quantified the discrimination of birdsongs by model neurons, examining both the dynamics and temporal resolution of discrimination, using a recently proposed spike distance metric (SDM). We found that single model neurons with delayed inhibitory STRFs can discriminate accurately between songs. Discrimination improves dramatically when the temporal structure of the neural response at fine timescales is considered. When we compared discrimination by model neurons with and without the inhibitory subregion, we found that the presence of the inhibitory subregion can improve discrimination. Finally, we modeled a cortical microcircuit with delayed synaptic inhibition, a candidate mechanism underlying delayed inhibitory STRFs, and showed that blocking inhibition in this model circuit degrades discrimination.

Coexisting chaotic attractors in a single neuron model with adapting feedback synapse

Coexisting chaotic attractors in a single neuron model with adapting feedback synapse

A Learning Rule to Model the Development of Orientation Selectivity in Visual Cortex

This paper presents a learning rule, CBA, to develop oriented receptive fields similar to those founded in cat striate cortex. The inherent complexity of the development of selectivity in visual cortex has led most authors to test their models by using a restricted input environment. Only recently, some learning rules (the PCA and the BCM rules) have been studied in a realistic visual environment. For these rules, which are based upon Hebbian learning, single neuron models have been proposed in order to get a better understanding of their properties and dynamics. These models suffered from unbounded growing of synaptic strength, which is remedied by a normalization process. However, normalization seems biologically implausible, given the non-local nature of this process. A detailed stability analysis of the proposed rule proves that the CBA attains a stable state without any need for normalization. Also, a comparison among the results achieved in different types of visual environments by the PCA, the BCM and the CBA rules is provided. The final results show that the CBA rule is appropriate for studying the biological process of receptive field formation and its application in image processing and artificial vision tasks.

Hopf bifurcation and chaos in a single inertial neuron model with time delay

A delayed differential equation modelling a single neuron with inertial term is considered in this paper. Hopf bifurcation is studied by using the normal form theory of retarded functional differential equations. When adopting a nonmonotonic activation function, chaotic behavior is observed. Phase plots, waveform plots, and power spectra are presented to confirm the chaoticity.

Coexisting chaotic attractors in a single neuron model with adapting feedback synapse

Abstract In this paper, we consider the nonlinear dynamical behavior of a single neuron model with adapting feedback synapse, and show that chaotic behaviors exist in this model. In some parameter domain, we observe two coexisting chaotic attractors, switching from the coexisting chaotic attractors to a connected chaotic attractor, and then switching back to the two coexisting chaotic attractors. We confirm the chaoticity by simulations with phase plots, waveform plots, and power spectra.

Classification capacity of a modular neural network implementing neurally inspired architecture and training rules.

A three-layer neural network (NN) with novel adaptive architecture has been developed. The hidden layer of the network consists of slabs of single neuron models, where neurons within a slab--but not between slabs--have the same type of activation function. The network activation functions in all three layers have adaptable parameters. The network was trained using a biologically inspired, guided-annealing learning rule on a variety of medical data. Good training/testing classification performance was obtained on all data sets tested. The performance achieved was comparable to that of SVM classifiers. It was shown that the adaptive network architecture, inspired from the modular organization often encountered in the mammalian cerebral cortex, can benefit classification performance.

An FPGA-Based Approach to High-Speed Simulation of Conductance-Based Neuron Models

The constant requirement for greater performance in neural model simulation has created the need for high-speed simulation platforms. We present a generalized, scalable field programmable gate array (FPGA)-based architecture for fast computation of neural models and focus on the steps involved in implementing a single-compartment and a two-compartment neuron model. Based on timing tests, it is shown that FPGAs can outperform traditional desktop computers in simulating these fairly simple models and would most likely provide even larger performance gains over computers in simulating more complex models. The potential of this method for improving neural modeling and dynamic clamping is discussed. In particular, it is believed that this approach could greatly speed up simulations of both highly complex single neuron models and networks of neurons. Additionally, our design is particularly well suited to automated parameter searches for tuning model behavior and to real-time simulation.

A Solution for the N-bit Parity Problem Using a Single Translated Multiplicative Neuron

A solution to the N-bit parity problem employing a single multiplicative neuron model, called translated multiplicative neuron (πt-neuron), is proposed. The πt-neuron presents the following advantages: (a) ∀N≥1, only 1 πt-neuron is necessary, with a threshold activation function and parameters defined within a specific interval; (b) no learning procedures are required; and (c) the computational cost is the same as the one associated with a simple McCulloch-Pitts neuron. Therefore, the πt-neuron solution to the N-bit parity problem has the lowest computational cost among the neural solutions presented to date.

Stochastic resonance and sensory information processing: a tutorial and review of application.

To review the stochastic resonance phenomena observed in sensory systems and to describe how a random process ('noise') added to a subthreshold stimulus can enhance sensory information processing and perception. Nonlinear systems need a threshold, subthreshold information bearing stimulus and 'noise' for stochastic resonance phenomena to occur. These three ingredients are ubiquitous in nature and man-made systems, which accounts for the observation of stochastic resonance in fields and conditions ranging from physics and engineering to biology and medicine. The stochastic resonance paradigm is compatible with single-neuron models or synaptic and channels properties and applies to neuronal assemblies activated by sensory inputs and perceptual processes as well. Here we review a few of the landmark experiments (including psychophysics, electrophysiology, fMRI, human vision, hearing and tactile functions, animal behavior, single/multiunit activity recordings). Models and experiments show a peculiar consistency with known neuronal and brain physiology. A number of naturally occurring 'noise' sources in the brain (e.g. synaptic transmission, channel gating, ion concentrations, membrane conductance) possibly accounting for stochastic resonance phenomena are also reviewed. Evidence is given suggesting a possible role of stochastic resonance in brain function, including detection of weak signals, synchronization and coherence among neuronal assemblies, phase resetting, 'carrier' signals, animal avoidance and feeding behaviors. Stochastic resonance is a ubiquitous and conspicuous phenomenon compatible with neural models and theories of brain function. The available evidence suggests cautious interpretation, but justifies research and should encourage neuroscientists and clinical neurophysiologists to explore stochastic resonance in biology and medical science.

Stability analysis of a single neuron model with delay

In this paper we study the asymptotic behavior and numerical approximation of the single neuron model equation x ̇ (t)=−dx(t)+af(x(t))+bf(x(t−τ))+I , t⩾0 (1), where d>0 and f( x)=0.5(| x+1|−| x−1|). We obtain new sufficient conditions for global asymptotic stability of constant equilibriums of (1), give several numerical examples to illustrate our results, and formulate conjectures on the asymptotic behavior of the solutions based on our numerical experiments.

Event-driven simulation of spiking neurons with stochastic dynamics.

We present a new technique, based on a proposed event-based strategy (Mattia & Del Giudice, 2000), for efficiently simulating large networks of simple model neurons. The strategy was based on the fact that interactions among neurons occur by means of events that are well localized in time (the action potentials) and relatively rare. In the interval between two of these events, the state variables associated with a model neuron or a synapse evolved deterministically and in a predictable way. Here, we extend the event-driven simulation strategy to the case in which the dynamics of the state variables in the inter-event intervals are stochastic. This extension captures both the situation in which the simulated neurons are inherently noisy and the case in which they are embedded in a very large network and receive a huge number of random synaptic inputs. We show how to effectively include the impact of large background populations into neuronal dynamics by means of the numerical evaluation of the statistical properties of single-model neurons under random current injection. The new simulation strategy allows the study of networks of interacting neurons with an arbitrary number of external afferents and inherent stochastic dynamics.

Computational modeling of bursting pacemaker neurons in the pre-Bötzinger complex

Bursting pacemaker neurons in the pre-Bötzinger complex (pBC) were modeled in the Hodgkin–Huxley style. The single neuron model included rapidly inactivating sodium, persistent sodium, and delayed-rectifier potassium currents. The kinetics of the rapidly inactivating and persistent sodium channels was modeled using experimental data obtained from whole-cell patch clamp recordings from pBC neurons in vitro. Our computational study focused on the conditions that could provide the generation of endogenous bursting activity in single pacemaker neurons and neural populations and on the specific roles of voltage-gated potassium and persistent sodium currents in triggering or suppression of endogenous population oscillations in the pBC.

Algorithmic analysis of irrational rotations in a single neuron model

We study computability of real-valued functions and the information needed for simulation of dynamical systems. In particular, we describe application of Kolmogorov complexity theory to computer simulation of irrational rotations in a single neuron model. We deduce the information needed for a parameter in simulating the dynamics by showing a difference in Kolmogorov complexity between a computable parameter and a non-computable parameter. Finally, we show that all trajectories generated by irrational rotations are non-computable iff its parameter is non-computable.

Bursting in Leech Heart Interneurons: Cell-Autonomous and Network-Based Mechanisms

Rhythmic activity within the heartbeat pattern generator of the medicinal leech is based on the alternating bursting of mutually inhibitory pairs of oscillator heart interneurons (half-center oscillators). Bicuculline methiodide has been shown to block mutual inhibition between these interneurons and to cause them to spike tonically while recorded intracellularly (Schmidt and Calabrese, 1992). Using extracellular recording techniques, we show here that oscillator and premotor heart interneurons continue to burst when pharmacologically isolated with bicuculline, although the bursting is not robust in some preparations. We propose that a nonspecific leak current introduced by the intracellular microelectrode suppresses endogenous bursting activity to account for the discrepancy with results using intracellular recording. A two-parameter bifurcation diagram (E(leak) vs g(leak)) of a mathematical model of a single heart interneuron shows a narrow stripe of parameter values where bursting occurs, separating large zones of tonic spiking and silence. A similar analysis performed for a half-center oscillator model outlined a much larger area of bursting. Bursting in the half-center oscillator model is also less sensitive to variation in the maximal conductances of voltage-gated currents than in the single-neuron model. Thus, in addition to ensuring appropriate bursting characteristics such as period, phase, and duty cycles, the half-center configuration enhances oscillation robustness, making them less susceptible to random or imposed changes in membrane parameters. Endogenous bursting, in turn, ensures appropriate bursting if the strength of mutual inhibition is weakened and limits the minimum period of the half-center oscillator to a period near that of the single neuron.

Resetting Intrinsic Purinergic Modulation of Neural Activity: An Associative Mechanism?

The purines, ATP and adenosine, control the rundown and termination of swimming in the Xenopus embryo. This intrinsic purinergic modulation, unavoidably present during every swimming episode, could lead to stereotyped inflexible behavior and consequently could jeopardize the survival of the embryo. To explore whether this control system can exhibit adaptability, I have used a minimal simulation in which a model neuron released ATP that (1) inhibited K+ currents and (2) was converted by ectonucleotidases to adenosine, which then inhibited Ca2+ currents. The model neuron exhibited an accommodating spike train controlled by the actions of ATP and adenosine. Feedforward inhibition by the upstream metabolite ADP of the ecto-5'-nucleotidase that converts AMP to adenosine introduced adaptability and allowed the resetting of spike accommodation. The strength of feedforward inhibition determined the extent to which resetting could occur. I have tested these predictions by examining swimming in the real embryo. The rundown of swimming was reset in a manner similar to that predicted by the single-neuron model. By blocking the purinoceptors, I have demonstrated that resetting in the embryo is attributable to the actions of the purines and results from feedforward inhibition of adenosine production. The resetting of rundown in the motor systems can be reformulated as an associative mechanism in which the temporal coincidence of two stimuli can prolong network activity if they fall within a particular time window. The length of the time window and the magnitude of the prolongation of neural activity both depend on the strength of the feedforward ADP-mediated inhibition of the ecto-5'-nucleotidase.

A modelling study on discrimination tasks

We investigate the discrimination capacity of a single neuron model including the integrate-and-fire (IF) model and the IF-FHN model [Neural Networks, 14 (2001) 955], both theoretically and numerically. Both magnitude and frequency discrimination tasks are considered. It is found that for the magnitude discrimination task, with a small fraction of coherent inputs, an IF model and an IF-FHN model with inhibitory inputs can tell one input from the other. The total probability of misclassifications (TPM) is considerably reduced with the increasing of inhibitory inputs. However, the IF model and the IF-FHN model are incapable of performing the frequency discrimination task.