Learning a Sparse Code for Temporal Sequences Using STDP and Sequence Compression

A spiking neural network that learns temporal sequences is described. A sparse code in which individual neurons represent sequences and subsequences enables multiple sequences to be stored without interference. The network is founded on a model of sequence compression in the hippocampus that is robust to variation in sequence element duration and well suited to learn sequences through spike-timing dependent plasticity (STDP). Three additions to the sequence compression model underlie the sparse representation: synapses connecting the neurons of the network that are subject to STDP, a competitive plasticity rule so that neurons specialize to individual sequences, and neural depolarization after spiking so that neurons have a memory. The response to new sequence elements is determined by the neurons that have responded to the previous subsequence, according to the competitively learned synaptic connections. Numerical simulations show that the model can learn sets of intersecting sequences, presented with widely differing frequencies, with elements of varying duration.

Exploring the tolerability of spatiotemporally complex electrical stimulation paradigms

A modified cortical stimulation model was used to investigate the effects of varying the synchronicity and periodicity of electrical stimuli delivered to multiple pairs of electrodes on seizure initiation. In this model, electrical stimulation of the motor cortex of rats, along four pairs of a microwire electrode array, results in an observable seizure with quantifiable electrographic duration and behavioural severity. Periodic stimuli had a constant inter-stimulus intervals across the two-second stimulus duration, whilst synchronous stimuli consisted of singular biphasic, bipolar pulses delivered to the four pairs of electrodes at precisely the same time for the entire two second stimulation period. In this way four combinations of stimulation were possible; periodic/synchronous (P/S), periodic/asynchronous (P/As), aperiodic/synchronous (Ap/S) and aperiodic/asynchronous (Ap/As). All stimulation types were designed with equal pulse width, current intensity and mean frequency of stimulation (60 Hz), standardizing net charge transfer. It was expected that the periodicity of the stimulus would be the primary determinant of seizure initiation and therefore severity and electrographic duration. However, the results showed that significant differences in both severity and duration only occurred when the synchronicity was altered. For periodic stimuli, synchronous delivery increased median seizure duration from 5 s to 13 s and increased median Racine severity from 1 to 3. In the aperiodic case, synchronous stimulus delivery increased median duration from 5.5 s to 11s and resulted in seizures of median severity 3 vs. 0 in the asynchronous case. These findings may have implications for the design of future neurostimulation waveform designs as higher numbers of electrodes and stimulator output channels become available in next generation implants.

Thermal heating of a retinal prosthesis: thermal model and in-vitro study

In order to develop retinal implants with a large number of electrodes, it is necessary to ensure that they do not cause damage to the neural tissue by the heat that the electrical circuits generate. Knowledge about the threshold of the amount of power that induces thermal damage will greatly assist in development of power budgets for implants, which has a significant effect upon the design of implant circuitry. In this study, we developed and tested in-vitro equipment that can dissipate thermal energy in current prosthesis implantation sites while simultaneously measuring and recording temperature distributions at multiple locations along the retinal tissue. A finite element thermal model of the feline eye was also created and validated by the in-vitro tests allowing for a much larger spectrum of thermal influences to be evaluated without the additional cost of animal sacrifice.

Differential stimulation of ON and OFF retinal ganglion cells: a modeling study

A model of the electrophysiological properties of ON and OFF retinal ganglion cells (RGCs) was constrained and validated using experimental data from the literature. Our simulations support experimental findings that differences in the magnitude of the T-type Ca(2+) current explain differences in the intrinsic electrophysiology of ON and OFF RGCs. The models are used to investigate the potential for differential stimulation of ON and OFF RGCs during neuroprosthetic stimulation with sinusoidal current. The model predicts that OFF cells fire preferential over ON cells in a frequency band around 10 Hz.

Theoretical framework for estimating the conductivity map of the retina through finite element analysis

A mathematical framework for estimation of the conductivity map of the retina is presented. The problem is formulated and solved in two-dimensional space considering hypothetical inhomogeneity in the conductivity profile at each layer of the retina in x and y directions. Finite element analysis is used to solve the equation of continuity in steady state to simulate voltage measurements as well as estimate the conductivity map. The results of simulated noisy data for an inhomogeneous retina layer and the fovea, which has a more complicated geometry, are presented. The error study of the estimated conductivity map shows that the error for an inhomogeneous conductivity profile is approximately 2% and the error for calculating the fovea conductivity map is just above 8%. This method can be extended to three-dimensions and can also be used to measure the impedance of different layers of the retina for alternating currents.

Simulating electrical stimulation of degenerative retinal ganglion cells with bi-phasic pulse trains

The aim of this work was to investigate how retinal ganglion cells (RGCs) respond to repetitive electrical stimulation in degenerative retina. The response of modeled ON and OFF cells was examined to bi-phasic pulse train stimulation of varying frequencies. Previously developed models of RGCs were extended to include an experimentally observable balance of excitatory and inhibitory currents in degenerative retina. The phenomena of fading and dark phosphenes with retinal implants were investigated. A hypothesis for a mechanism contributing to these phenomena was formulated.

Viability of the inner retina in a novel mouse model of retinitis pigmentosa

Retinal prostheses aim to restore vision to patients who are blind from photoreceptor diseases such as Retinitis Pigmentosa (RP). All implants target the neural cells in the inner retina, the retinal ganglion cells (RGCs). Our research focuses on further understanding the disease process of RP during mid to late stages when total loss of photoreceptors has occurred and significant remodeling of inner retinal neurons has taken place. We have used a novel transgenic mouse, Rd1-FTL, to observe different degenerative stages of RP. Notably, in the aged retina we have evidence that there was gross inner retinal remodeling as well as glial dysfunction that occurred in confined regions in the central retina that worsened overtime. Consequently, the timing of implantation and location of the prosthesis both need to account for the state of the retina at different stages in the disease process.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks V: self-organization schemes and weight dependence

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity on a (much) slower time scale. This paper examines the effect of STDP in a recurrently connected network stimulated by external pools of input spike trains, where both input and recurrent synapses are plastic. Our previously developed theoretical framework is extended to incorporate weight-dependent STDP and dendritic delays. The weight dynamics is determined by an interplay between the neuronal activation mechanisms, the input spike-time correlations, and the learning parameters. For the case of two external input pools, the resulting learning scheme can exhibit a symmetry breaking of the input connections such that two neuronal groups emerge, each specialized to one input pool only. In addition, we show how the recurrent connections within each neuronal group can be strengthened by STDP at the expense of those between the two groups. This neuronal self-organization can be seen as a basic dynamical ingredient for the emergence of neuronal maps induced by activity-dependent plasticity.

Representation of input structure in synaptic weights by spike-timing-dependent plasticity

Spike-timing-dependent plasticity (STDP) has been shown to generate a synaptic weight structure that is determined by the timing of the pre- and postsynaptic spikes at the synapse. In this paper it is shown under what conditions a neuron stimulated by several pools of delta-correlated inputs encodes this input structure in its resulting weight structure. The analysis is carried out using Poisson neurons with weight-dependent STDP. The learning dynamics induced by STDP leads to both stabilization of the input weights and competition between the weights for a broad range of learning parameters. The results demonstrate how weight-dependent STDP can generate multimodal stable asymptotic distributions of the synaptic weights.

Seizure severity and duration in the cortical stimulation model of experimental epilepsy in rats: a longitudinal study

The aim of this study was to determine the current intensities necessary to elicit three levels of varying EEG and behavioural phenomena with electrical stimulation, and also to determine the consistency of the EEG and behavioural components of the triggered seizures over time. Electrical stimulation of the primary motor/somatosensory cortex was performed in 16 adult rats with multichannel microwire electrode arrays. Stimulation was delivered at a frequency of 60 Hz (1 ms pulse width), for 2 s duration, as biphasic rectangular pulses over four of the eight available electrode pairs. Current intensity thresholds for interruption of normal behaviour, epileptiform afterdischarge (EAD) longer than 5 s and motor seizures with Racine severity greater than 3 were not correlated to time post-surgery. The Racine threshold was shown to be negatively correlated to the EAD duration and Racine severity of seizures elicited in the following sessions. Seizures were reliably generated in rats through cortical stimulation with microwire electrode arrays and these seizures were not shown to be subject to any kindling type effects up to 53 days post-implantation. Both the electrographic duration and behavioural severity of stimulated seizures remained, on average, constant during this experimental period. Approximately one-third of stimulations did not cause observable motor seizures and of those that did result in seizures, forelimb clonus was the most common manifestation and the mean EAD duration was 18.5 s. No damage beyond that caused by surgical implantation of electrodes was observed in the histological analyses of stimulated and non-stimulated tissue. The consistency, duration and severity of seizures within this timeframe make this cortical stimulation model suitable for investigations into novel therapeutic interventions for epilepsy that require a known seizure focus.

Spiking Neuron Model for Temporal Sequence Recognition

A biologically inspired neuronal network that stores and recognizes temporal sequences of symbols is described. Each symbol is represented by excitatory input to distinct groups of neurons (symbol pools). Unambiguous storage of multiple sequences with common subsequences is ensured by partitioning each symbol pool into subpools that respond only when the current symbol has been preceded by a particular sequence of symbols. We describe synaptic structure and neural dynamics that permit the selective activation of subpools by the correct sequence. Symbols may have varying durations of the order of hundreds of milliseconds. Physiologically plausible plasticity mechanisms operate on a time scale of tens of milliseconds; an interaction of the excitatory input with periodic global inhibition bridges this gap so that neural events representing successive symbols occur on this much faster timescale. The network is shown to store multiple overlapping sequences of events. It is robust to variation in symbol duration, it is scalable, and its performance degrades gracefully with perturbation of its parameters.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks IV: structuring synaptic pathways among recurrent connections

In neuronal networks, the changes of synaptic strength (or weight) performed by spike-timing-dependent plasticity (STDP) are hypothesized to give rise to functional network structure. This article investigates how this phenomenon occurs for the excitatory recurrent connections of a network with fixed input weights that is stimulated by external spike trains. We develop a theoretical framework based on the Poisson neuron model to analyze the interplay between the neuronal activity (firing rates and the spike-time correlations) and the learning dynamics, when the network is stimulated by correlated pools of homogeneous Poisson spike trains. STDP can lead to both a stabilization of all the neuron firing rates (homeostatic equilibrium) and a robust weight specialization. The pattern of specialization for the recurrent weights is determined by a relationship between the input firing-rate and correlation structures, the network topology, the STDP parameters and the synaptic response properties. We find conditions for feed-forward pathways or areas with strengthened self-feedback to emerge in an initially homogeneous recurrent network.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks III: Partially connected neurons driven by spontaneous activity

In contrast to a feed-forward architecture, the weight dynamics induced by spike-timing-dependent plasticity (STDP) in a recurrent neuronal network is not yet well understood. In this article, we extend a previous study of the impact of additive STDP in a recurrent network that is driven by spontaneous activity (no external stimulating inputs) from a fully connected network to one that is only partially connected. The asymptotic state of the network is analyzed, and it is found that the equilibrium and stability conditions for the firing rates are similar for both full and partial connectivity: STDP causes the firing rates to converge toward the same value and remain quasi-homogeneous. However, when STDP induces strong weight competition, the connectivity affects the weight dynamics in that the distribution of the weights disperses more quickly for lower density than for higher density. The asymptotic weight distribution strongly depends upon that at the beginning of the learning epoch; consequently, homogeneous connectivity alone is not sufficient to obtain homogeneous neuronal activity. In the absence of external inputs, STDP can nevertheless generate structure in the network through autocorrelation effects, for example, by introducing asymmetry in network topology.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. II. Input selectivity--symmetry breaking

Spike-timing-dependent plasticity (STDP) is believed to structure neuronal networks by slowly changing the strengths (or weights) of the synaptic connections between neurons depending upon their spiking activity, which in turn modifies the neuronal firing dynamics. In this paper, we investigate the change in synaptic weights induced by STDP in a recurrently connected network in which the input weights are plastic but the recurrent weights are fixed. The inputs are divided into two pools with identical constant firing rates and equal within-pool spike-time correlations, but with no between-pool correlations. Our analysis uses the Poisson neuron model in order to predict the evolution of the input synaptic weights and focuses on the asymptotic weight distribution that emerges due to STDP. The learning dynamics induces a symmetry breaking for the individual neurons, namely for sufficiently strong within-pool spike-time correlation each neuron specializes to one of the input pools. We show that the presence of fixed excitatory recurrent connections between neurons induces a group symmetry-breaking effect, in which neurons tend to specialize to the same input pool. Consequently STDP generates a functional structure on the input connections of the network.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. I. Input selectivity--strengthening correlated input pathways

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity. In this paper, we extend previous studies of input selectivity induced by (STDP) for single neurons to the biologically interesting case of a neuronal network with fixed recurrent connections and plastic connections from external pools of input neurons. We use a theoretical framework based on the Poisson neuron model to analytically describe the network dynamics (firing rates and spike-time correlations) and thus the evolution of the synaptic weights. This framework incorporates the time course of the post-synaptic potentials and synaptic delays. Our analysis focuses on the asymptotic states of a network stimulated by two homogeneous pools of "steady" inputs, namely Poisson spike trains which have fixed firing rates and spike-time correlations. The (STDP) model extends rate-based learning in that it can implement, at the same time, both a stabilization of the individual neuron firing rates and a slower weight specialization depending on the input spike-time correlations. When one input pathway has stronger within-pool correlations, the resulting synaptic dynamics induced by (STDP) are shown to be similar to those arising in the case of a purely feed-forward network: the weights from the more correlated inputs are potentiated at the expense of the remaining input connections.

The thalamocortical circuit and the generation of epileptic spikes in rat models of focal epilepsy

We investigate thalamocortical interactions in the tetanus toxin and the cortical stimulation rat models of epilepsy. Using local field potential recordings from the cortex and the thalamus of the rat, the nonlinear regression index is calculated to create the direction index in order to study neurodynamics during seizures. Coarse time-scale analysis reveals that the cortex drives the thalamus for the majority of the time during seizures. However, fine time-scale analysis provides evidence that epileptic spikes are driven from the thalamus. This new result has implications for understanding, diagnosing and using electrical stimulation to treat epileptic seizures.

Spike-timing-dependent plasticity for neurons with recurrent connections

The dynamics of the learning equation, which describes the evolution of the synaptic weights, is derived in the situation where the network contains recurrent connections. The derivation is carried out for the Poisson neuron model. The spiking-rates of the recurrently connected neurons and their cross-correlations are determined self- consistently as a function of the external synaptic inputs. The solution of the learning equation is illustrated by the analysis of the particular case in which there is no external synaptic input. The general learning equation and the fixed-point structure of its solutions is discussed.

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker-Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).

A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input

The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker-Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin-Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to - the irregular nature of spiking in cortical neurons and neural gain modulation.

Learning the structure of correlated synaptic subgroups using stable and competitive spike-timing-dependent plasticity

Synaptic plasticity must be both competitive and stable if ongoing learning of the structure of neural inputs is to occur. In this paper, a wide class of spike-timing-dependent plasticity (STDP) models is identified that have both of these desirable properties in the case in which the input consists of subgroups of synapses that are correlated within the subgroup through the occurrence of simultaneous input spikes. The process of synaptic structure formation is studied, illustrating one particular class of these models. When the learning rate is small, multiple alternative synaptic structures are possible given the same inputs, with the outcome depending on the initial weight configuration. For large learning rates, the synaptic structure does not stabilize, resulting in neurons without consistent response properties. For learning rates in between, a unique and stable synaptic structure typically forms. When this synaptic structure exhibits a bimodal distribution, the neuron will respond selectively to one or more of the subgroups. The robustness with which this selectivity develops during learning is largely determined by the ratio of the subgroup correlation strength to the number of subgroups. The fraction of potentiated subgroups is primarily determined by the balance between potentiation and depression.

Spike-Timing-Dependent Plasticity: The Relationship to Rate-Based Learning for Models with Weight Dynamics Determined by a Stable Fixed Point

Experimental evidence indicates that synaptic modification depends on the timing relationship between the presynaptic inputs and the output spikes that they generate. In this letter, results are presented for models of spike-timing-dependent plasticity (STDP) whose weight dynamics is determined by a stable fixed point. Four classes of STDP are identified on the basis of the time extent of their input-output interactions. The effect on the potentiation of synapses with different rates of input is investigated to elucidate the relationship of STDP with classical studies of long-term potentiation and depression and rate-based Hebbian learning. The selective potentiation of higher-rate synaptic inputs is found only for models where the time extent of the input-output interactions is input restricted (i.e., restricted to time domains delimited by adjacent synaptic inputs) and that have a time-asymmetric learning window with a longer time constant for depression than for potentiation. The analysis provides an account of learning dynamics determined by an input-selective stable fixed point. The effect of suppressive interspike interactions on STDP is also analyzed and shown to modify the synaptic dynamics.

An analytical model for the "large, fluctuating synaptic conductance state" typical of neocortical neurons in vivo

A model of in vivo-like neocortical activity is studied analytically in relation to experimental data and other models in order to understand the essential mechanisms underlying such activity. The model consists of a network of sparsely connected excitatory and inhibitory integrate-and-fire (IF) neurons with conductance-based synapses. It is shown that the model produces values for five quantities characterizing in vivo activity that are in agreement with both experimental ranges and a computer-simulated Hodgkin-Huxley model adapted from the literature (Destexhe et al. (2001) Neurosci. 107(1): 13-24). The analytical model builds on a study by Brunel (2000) (J. Comput. Neurosci. 8: 183-208), which used IF neurons with current-based synapses, and therefore does not account for the full range of experimental data. The present results suggest that the essential mechanism required to explain a range of data on in vivo neocortical activity is the conductance-based synapse and that the particular model of spike initiation used is not crucial. Thus the IF model with conductance-based synapses may provide a basis for the analytical study of the "large, fluctuating synaptic conductance state" typical of neocortical neurons in vivo.

Ventral cochlear nucleus coding of voice onset time in naturally spoken syllables

These experiments examined the coding of the voice onset time (VOT) of six naturally spoken syllables, presented at a number of intensities, by ventral cochlear nucleus (VCN) neurons in rats anesthetized with urethane. VOT is one of the cues for the identification of a stop consonant, and is defined by the interval between stop release and the first glottal pulse that marks the onset of voicing associated with a vowel. The syllables presented (/bot/, /dot/, /got/, /pot/, /tot/, /kot/) each had a different VOT, ranging between 10 and 108 ms. Extracellular recordings were made from single neurons (N=202) with a wide range of best frequencies (BFs; 0.66-10 kHz) that represented the major VCN response types - primary-like (67.8% of sample), chopper (19.8%), and onset (12.4%) neurons. The different VOTs of the syllables were accurately reflected in sharp, precisely timed, and statistically significant changes in average discharge rate in all cell types, as well as the entire VCN sample. The prominence of the response to stop release and voice onset, and the level of activity prior to the VOT, were influenced by syllable intensity and the spectrum of stop release, as well as cell BF and type. Our results suggest that the responses of VCN cells with BFs above the first formant frequency are dominated by their sensitivity to the onsets of broadband events in speech, and allows them to convey accurate information about a syllable's VOT.

How synapses in the auditory system wax and wane: theoretical perspectives

Spike-timing-dependent synaptic plasticity has recently provided an account of both the acuity of sound localization and the development of temporal-feature maps in the avian auditory system. The dynamics of the resulting learning equation, which describes the evolution of the synaptic weights, is governed by an unstable fixed point. We outline the derivation of the learning equation for both the Poisson neuron model and the leaky integrate-and-fire neuron with conductance synapses. The asymptotic solutions of the learning equation can be described by a spectral representation based on a biorthogonal expansion.

Study of neuronal gain in a conductance-based leaky integrate-and-fire neuron model with balanced excitatory and inhibitory synaptic input

Neurons receive a continual stream of excitatory and inhibitory synaptic inputs. A conductance-based neuron model is used to investigate how the balanced component of this input modulates the amplitude of neuronal responses. The output spiking rate is well described by a formula involving three parameters: the mean mu and variance sigma of the membrane potential and the effective membrane time constant tauQ. This expression shows that, for sufficiently small tauQ, the level of balanced excitatory-inhibitory input has a nonlinear modulatory effect on the neuronal gain.

Summation of spatiotemporal input patterns in leaky integrate-and-fire neurons: application to neurons in the cochlear nucleus receiving converging auditory nerve fiber input

The response of leaky integrate-and-fire neurons is analyzed for periodic inputs whose phases vary with their spatial location. The model gives the relationship between the spatial summation distance and the degree of phase locking of the output spikes (i.e., locking to the periodic stochastic inputs, measured by the synchronization index). The synaptic inputs are modeled as an inhomogeneous Poisson process, and the analysis is carried out in the Gaussian approximation. The model has been applied to globular bushy cells of the cochlear nucleus, which receive converging inputs from auditory nerve fibers that originate at neighboring sites in the cochlea. The model elucidates the roles played by spatial summation and coincidence detection, showing how synchronization decreases with an increase in both frequency and spatial spread of inputs. It also shows under what conditions an enhancement of synchronization of the output relative to the input takes place.

Synchronization of the neural response to noisy periodic synaptic input

The timing information contained in the response of a neuron to noisy periodic synaptic input is analyzed for the leaky integrate-and-fire neural model. We address the question of the relationship between the timing of the synaptic inputs and the output spikes. This requires an analysis of the interspike interval distribution of the output spikes, which is obtained in the gaussian approximation. The conditional output spike density in response to noisy periodic input is evaluated as a function of the initial phase of the inputs. This enables the phase transition matrix to be calculated, which relates the phase at which the output spike is generated to the initial phase of the inputs. The interspike interval histogram and the period histogram for the neural response to ongoing periodic input are then evaluated by using the leading eigenvector of this phase transition matrix. The synchronization index of the output spikes is found to increase sharply as the inputs become synchronized. This enhancement of synchronization is most pronounced for large numbers of inputs and lower frequencies of modulation and also for rates of input near the critical input rate. However, the mutual information between the input phase of the stimulus and the timing of output spikes is found to decrease at low input rates as the number of inputs increases. The results show close agreement with those obtained from numerical simulations for large numbers of inputs.

Balanced neurons: analysis of leaky integrate-and-fire neurons with reversal potentials

A new technique is presented for analyzing leaky integrate-and-fire neurons that incorporates reversal potentials, which impose a biologically realistic lower bound to the membrane potential. The time distribution of the synaptic inputs is modeled as a Poisson process. The analysis is carried out in the Gaussian approximation, which comparison with numerical simulations confirms is most accurate in the limit of a large number of inputs. The hypothesis that the observed variability in the spike times of cortical neurons is caused by a balance of excitatory and inhibitory synaptic inputs is supported by the results for the coefficient of variation of the interspike intervals. Its value decreases with both increasing numbers and amplitude of inputs, and is consistently lower than 1.0 over a wide range of realistic parameter values. The dependence of the output spike rate upon the rate, number, and amplitude of the synaptic inputs, as well as upon the value of the inhibitory reversal potential, is given.

Delay analysis in the auditory brainstem of the rat: comparison with click latency

Many cells in the auditory brainstem 'phase lock' to tone stimuli. From the changing phase relationship between the stimulus and the neural response in phase-locking cells, the delay between them can be estimated. This delay, however, is consistently greater than the latency measured in response to click stimuli, an important discrepancy. In this paper the different measures of delay, namely phase delay, group delay and signal-front delay are re-examined. An improved method for computing the average group delay is presented, which accounts for the cyclical nature of the phase data. Data were collected from units in successive processing sites of auditory pathway: the auditory nerve, the cochlear nucleus, the trapezoid body and the medial nucleus of the trapezoid body. Low-characteristic frequency (CF) units gave multimodal post-stimulus-time histograms in response to clicks, and showed stepwise decreases in latency with increasing intensity, with the appearance of earlier peaks in the response, rather than shifts in the timing of the peaks. The separation of peaks corresponded to the inverse of the unit's CF. High-CF units also showed a decline in click latency with intensity, but to a lesser degree than low CF units. We present an analysis which explains the difference between click latency and delay, and which in contrast to previous accounts is experimentally testable. We demonstrate that this new framework accounts for the discrepancy between the two measures of delay, and in addition accounts for the observed stepwise shifts in click latency for low-CF units.

Temporal processing from the auditory nerve to the medial nucleus of the trapezoid body in the rat

This investigation examines temporal processing through successive sites in the rat auditory pathway: auditory nerve (AN), anteroventral cochlear nucleus (AVCN) and the medial nucleus of the trapezoid body (MNTB). The degree of phase-locking, measured as vector strength, varied with intensity relative to the cell's threshold, and saturated at a value that depended upon stimulus frequency. A typical pattern showed decline in the saturated vector strength from approximately 0.8 at 400 Hz to about 0.3 at 2000 Hz, with similar profiles in units with a range of characteristic frequencies (480-32,000 Hz). A new expression for temporal dispersion indicates that this variation corresponds to a limiting degree of temporal imprecision, which is relatively consistent between different cells. From AN to AVCN, an increase in vector strength was seen for frequencies below 1000 Hz. At higher frequencies, a decrease in vector strength was observed. From AVCN to MNTB a tendency for temporal coding to be improved below 800 Hz and degraded further above 1500 Hz was seen. This change in temporal processing ability could be attributed to units classified as primary-like with notch (PL(N)). PL(N) MNTB units showed a similar vector strength distribution to PL(N) AVCN units. Our results suggest that AVCN PL(N) units, representing globular bushy cells, are specialised for enhancing the temporal code at low frequencies and relaying this information to principal cells of the MNTB.

Shot noise in the leaky integrate-and-fire neuron

We study the influence of noise on the transmission of temporal information by a leaky integrate-and-fire neuron using the theory of shot noise. The model includes a finite number of synapses and has a membrane potential variance de facto modulated by the input signal. The phenomenon of stochastic resonance in spiking neurons is analytically exhibited using an inhomogeneous Poisson process model of the spike trains, and links with the traditional Ornstein-Uhlenbeck process obtained by a diffusion approximation are given. It is shown that the modulated membrane potential variance inherent to the model gives better signal processing capabilities than the diffusion approximation.

Calculation of interspike intervals for integrate-and-fire neurons with poisson distribution of synaptic inputs

We present a new technique for calculating the interspike intervals of integrate-and-fire neurons. There are two new components to this technique. First, the probability density of the summed potential is calculated by integrating over the distribution of arrival times of the afferent post-synaptic potentials (PSPs), rather than using conventional stochastic differential equation techniques. A general formulation of this technique is given in terms of the probability distribution of the inputs and the time course of the postsynaptic response. The expressions are evaluated in the gaussian approximation, which gives results that become more accurate for large numbers of small-amplitude PSPs. Second, the probability density of output spikes, which are generated when the potential reaches threshold, is given in terms of an integral involving a conditional probability density. This expression is a generalization of the renewal equation, but it holds for both leaky neurons and situations in which there is no time-translational invariance. The conditional probability density of the potential is calculated using the same technique of integrating over the distribution of arrival times of the afferent PSPs. For inputs with a Poisson distribution, the known analytic solutions for both the perfect integrator model and the Stein model (which incorporates membrane potential leakage) in the diffusion limit are obtained. The interspike interval distribution may also be calculated numerically for models that incorporate both membrane potential leakage and a finite rise time of the postsynaptic response. Plots of the relationship between input and output firing rates, as well as the coefficient of variation, are given, and inputs with varying rates and amplitudes, including inhibitory inputs, are analyzed. The results indicate that neurons functioning near their critical threshold, where the inputs are just sufficient to cause firing, display a large variability in their spike timings.

Analysis of integrate-and-fire neurons: synchronization of synaptic input and spike output

A new technique for analyzing the probability distribution of output spikes for the integrate-and-fire model is presented. This technique enables us to investigate models with arbitrary synaptic response functions that incorporate both leakage across the membrane and a rise time of the postsynaptic potential. The results, which are compared with numerical simulations, are exact in the limit of a large number of small-amplitude inputs. This method is applied to the synchronization problem, in which we examine the relationship between the spread in arrival times of the inputs (the temporal jitter of the synaptic input) and the resultant spread in the times at which the output spikes are generated (output jitter). The results of previous studies, which indicated that the ration of the output jitter to the input jitter is consistently less than one and that it decreases for increasing numbers of inputs, are confirmed for three classes of the integrate-and-fire model. In addition to the previously identified factors of axonal propagation times and synaptic jitter, we identify the variation in the spike-generating thresholds of the neurons and the variation in the number of active inputs as being important factors that determine the timing jitter in layered networks. Previously observed phase differences between optimally and suboptimally stimulated neurons may be understood in terms of the relative time taken to reach threshold.

Intracellular responses of the rat cochlear nucleus to sound and its role in temporal coding

The anteroventral cochlear nucleus (AVCN), the first centre of the central auditory pathway, contains globular bushy cells, which are unique in their ability to produce fast excitatory post-synaptic potentials (EPSPs). Using in vivo intracellular recordings in the rat AVCN we examined these fast EPSPs in relation to temporal coding. At frequencies up to 2.5 kHz, EPSPs were evoked on successive sine waves of the stimulus with EPSP summation limited. This one-to-one relationship between the EPSPs and the sound wave period was present at higher frequencies and over a greater intensity range than for action potentials. These results suggest that temporal coding is possible in globular bushy neurones by their ability to extract temporal information through fast processing of convergent presynaptic input.

The Facial Musculature of the Australian Aboriginal: Part II

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An Abnormal Skull from New Guinea, with Remarks on the Structure of the Mandible

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Preliminary Observations on the Nose of the Australian Aboriginal, with a Table of Aboriginal Head Measurements

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