Retinal ganglion cells: mechanisms underlying depolarization block and differential responses to high frequency electrical stimulation of ON and OFF cells.

OBJECTIVE: ON and OFF retinal ganglion cells (RGCs) are known to have non-monotonic responses to increasing amplitudes of high frequency (2 kHz) biphasic electrical stimulation. That is, an increase in stimulation amplitude causes an increase in the cell's spike rate up to a peak value above which further increases in stimulation amplitude cause the cell to decrease its activity. The peak response for ON and OFF cells occurs at different stimulation amplitudes, which allows differential stimulation of these functional cell types. In this study, we investigate the mechanisms underlying the non-monotonic responses of ON and OFF brisk-transient RGCs and the mechanisms underlying their differential responses. APPROACH: Using in vitro patch-clamp recordings from rat RGCs, together with simulations of single and multiple compartment Hodgkin-Huxley models, we show that the non-monotonic response to increasing amplitudes of stimulation is due to depolarization block, a change in the membrane potential that prevents the cell from generating action potentials. MAIN RESULTS: We show that the onset for depolarization block depends on the amplitude and frequency of stimulation and reveal the biophysical mechanisms that lead to depolarization block during high frequency stimulation. Our results indicate that differences in transmembrane potassium conductance lead to shifts of the stimulus currents that generate peak spike rates, suggesting that the differential responses of ON and OFF cells may be due to differences in the expression of this current type. We also show that the length of the axon's high sodium channel band (SOCB) affects non-monotonic responses and the stimulation amplitude that leads to the peak spike rate, suggesting that the length of the SOCB is shorter in ON cells. SIGNIFICANCE: This may have important implications for stimulation strategies in visual prostheses.

Optimized single pulse stimulation strategy for retinal implants.

Retinal implants offer prospects of vision restoration for some blind patients by eliciting visual percepts of spots of light called 'phosphenes'. Recently, a mathematical model has been developed that predicts patients' perception of phosphene brightness for current-driven electrical stimulation of the retina. This model is explored for different stimulation parameters on a single electrode, including safety and hardware limitations, to produce phosphenes of specified brightness. We describe a procedure to derive stimulation parameters to account for such constraints, and describe methods to construct optimal stimuli in terms of producing maximal perceived brightness and efficient generation of phosphenes of a given brightness by employing minimal energy. In both cases, it is found that the resulting optimized stimulation waveforms consist of a long stimulation period, and interphase delays between initial and charge-balancing phases.

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).

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.

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.

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.

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.