Sparse coding, predictive coding and divisive normalization have each been found to be principles that underlie the function of neural circuits in many parts of the brain, supported by substantial experimental evidence. However, the connections between these related principles are still poorly understood. Sparse coding and predictive coding can be reconciled into a learning framework with predictive structure and sparse responses, termed as sparse/predictive coding. However, how sparse/predictive coding (a learning model) is connected with divisive normalization (not a learning model) is still not well investigated. In this paper, we show how sparse coding, predictive coding, and divisive normalization can be described within a unified framework, and illustrate this explicitly within the context of a two-layer neural learning model of sparse/predictive coding. This two-layer model is constructed in a way that implements sparse coding with a network structure that is constructed by implementing predictive coding. We demonstrate how a homeostatic function that regulates neural responses in the model can shape the nonlinearity of neural responses in a way that replicates different forms of divisive normalization. Simulations show that the model can learn simple cells in the primary visual cortex with the property of contrast saturation, which has previously been explained by divisive normalization. In summary, the study demonstrates that the three principles of sparse coding, predictive coding, and divisive normalization can be connected to provide a learning framework based on biophysical properties, such as Hebbian learning and homeostasis, and this framework incorporates both learning and more diverse response nonlinearities observed experimentally. This framework has the potential to also be used to explain how the brain learns to integrate input from different sensory modalities.

This study introduces a neurobiologically inspired computational model based on the predictive coding algorithm, providing insights into coherent motion detection processes. The model is designed to reflect key principles observed in the visual system, particularly MT neurons and their surround suppression mechanisms, which play a critical role in detecting global motion. By integrating these principles, the model simulates how motion structures are decomposed into individual and shared sources, mirroring the brain's strategy for extracting coherent motion patterns. The results obtained from random dot stimuli underscore the delicate balance between sensory data and prior knowledge in motion detection. Model testing across varying noise levels reveals that, as noise increases, the model takes longer to stabilize its motion estimates, consistent with psychophysical experiments showing that response duration (e.g., reaction time or decision-making time) also increases under higher noise conditions. The model suggests that an excessive emphasis on prior knowledge prolongs the stabilization time for motion detection, whereas an optimal integration of prior expectations enhances detection accuracy and efficiency by preventing excessive disturbances due to noise. These findings contribute to potential explanations for motion detection deficiencies observed in schizophrenia.

Many experimental and computational studies deal with sparseness, balance, and decorrelation in neural networks and explain the presence of these properties as fulfilling requirements related to optimum energy efficiency, network stability, and information representation. These studies leave the question of how these properties arise in the brain unanswered. The present study attempts to address this question using a model built upon the experimentally observed properties of neural responses, homeostasis, and synaptic plasticity. The experimentally observed properties of sparseness, balance, and decorrelation are then expected to emerge from this substrate. A spiking neural model of the primary visual cortex (V1) was investigated. Populations of both inhibitory and excitatory leaky integrate-and-fire neurons with recurrent connections were provided with spiking input from simulated ON and OFF neurons of the lateral geniculate nucleus. This network was provided with natural image stimuli as input. All synapses underwent learning using spike-timing-dependent plasticity learning rules. A homeostatic rule adjusted the weights and thresholds of each neuron based on target homeostatic spiking rates and mean synaptic input values. These experimentally grounded rules resulted in a number of the expected properties of information representation. The network showed a temporally sparse spike response to inputs and this was associated with a sparse code with Gabor-like receptive fields. The network was balanced at both slow and fast time scales; increased excitatory input was balanced by increased inhibition. This balance was associated with decorrelated firing that was observed as population sparseness. This population sparseness was both the cause and result of the decorrelation of receptive fields. These observed emergent properties (balance, temporal sparseness, population sparseness, and decorrelation) indicate that the network is implementing expected principles of information processing: efficient coding, information maximization ('infomax'), and a lateral or single-layer form of predictive coding. These emergent features of the network were shown to be robust to randomized jitter of the values of key simulation parameters.

This study focuses on the use of a neural mass model to investigate potential relationships between functional connectivity and seizure frequency in epilepsy. We fitted a three-layer neural mass model of a cortical column to intracranial EEG (iEEG) data from a Tetanus Toxin rat model of epilepsy, which also included responses to periodic electrical stimulation. Our results show that some of the connectivity weights between different neural populations correlate significantly with the number of seizures each day, offering valuable insights into the dynamics of neural circuits during epileptogenesis. We also simulated single-pulse electrical stimulation of the neuronal populations to observe their responses after the connectivity weights were optimized to fit background (non-seizure) EEG data. The recovery time, defined as the time from stimulation until the membrane potential returns to baseline, was measured as a representation of the critical slowing down phenomenon observed in nonlinear systems operating near a bifurcation boundary. The results revealed that recovery times in the responses of the computational model fitted to the EEG data were longer during 5 min periods preceding seizures compared to 1 hr before seizures in four out of six rats. Analysis of the iEEG recorded in response to electrical stimulation revealed results similar to the computational model in four out of six rats. This study supports the potential use of this computational model as a model-based biomarker for seizure prediction when direct electrical stimulation to the brain is not feasible.

Strong inhibitory recurrent connections can reduce the tendency for a neural network to become unstable. This is known as inhibitory stabilization; networks that are unstable in the absence of strong inhibitory feedback because of their unstable excitatory recurrent connections are known as Inhibition Stabilized Networks (ISNs). One of the characteristics of ISNs is their "paradoxical response", where perturbing the inhibitory neurons with additional excitatory input results in a decrease in their activity after a temporal delay instead of increasing their activity. Here, we develop a model of populations of neurons across different layers of cortex. Within each layer, there is one population of inhibitory neurons and one population of excitatory neurons. The connectivity weights across different populations in the model are derived from a synaptic physiology database provided by the Allen Institute. The model shows a gradient of excitation-inhibition balance across different layers in the cortex, where superficial layers are more inhibitory dominated compared to deeper layers. To investigate the presence of ISNs across different layers, we measured the membrane potentials of neural populations in the model after perturbing inhibitory populations. The results show that layer 2/3 in the model does not operate in the ISN regime but layers 4 and 5 do operate in the ISN regime. These results accord with neurophysiological findings that explored the presence of ISNs across different layers in the cortex. The results show that there may be a systematic macroscopic gradient of inhibitory stabilization across different layers in the cortex that depends on the level of excitation-inhibition balance, and that the strength of the paradoxical response increases as the model moves closer to bifurcation points.

The ability of the brain to represent the external world in real-time is impacted by the fact that neural processing takes time. Because neural delays accumulate as information progresses through the visual system, representations encoded at each hierarchical level are based upon input that is progressively outdated with respect to the external world. This 'representational lag' is particularly relevant to the task of localizing a moving object-because the object's location changes with time, neural representations of its location potentially lag behind its true location. Converging evidence suggests that the brain has evolved mechanisms that allow it to compensate for its inherent delays by extrapolating the position of moving objects along their trajectory. We have previously shown how spike-timing dependent plasticity (STDP) can achieve motion extrapolation in a two-layer, feedforward network of velocity-tuned neurons, by shifting the receptive fields of second layer neurons in the opposite direction to a moving stimulus. The current study extends this work by implementing two important changes to the network to bring it more into line with biology: we expanded the network to multiple layers to reflect the depth of the visual hierarchy, and we implemented more realistic synaptic time-courses. We investigate the accumulation of STDP-driven receptive field shifts across several layers, observing a velocity-dependent reduction in representational lag. These results highlight the role of STDP, operating purely along the feedforward pathway, as a developmental strategy for delay compensation.

There are two distinct classes of cells in the primary visual cortex (V1): simple cells and complex cells. One defining feature of complex cells is their spatial phase invariance; they respond strongly to oriented grating stimuli with a preferred orientation but with a wide range of spatial phases. A classical model of complete spatial phase invariance in complex cells is the energy model, in which the responses are the sum of the squared outputs of two linear spatially phase-shifted filters. However, recent experimental studies have shown that complex cells have a diverse range of spatial phase invariance and only a subset can be characterized by the energy model. While several models have been proposed to explain how complex cells could learn to be selective to orientation but invariant to spatial phase, most existing models overlook many biologically important details. We propose a biologically plausible model for complex cells that learns to pool inputs from simple cells based on the presentation of natural scene stimuli. The model is a three-layer network with rate-based neurons that describes the activities of LGN cells (layer 1), V1 simple cells (layer 2), and V1 complex cells (layer 3). The first two layers implement a recently proposed simple cell model that is biologically plausible and accounts for many experimental phenomena. The neural dynamics of the complex cells is modeled as the integration of simple cells inputs along with response normalization. Connections between LGN and simple cells are learned using Hebbian and anti-Hebbian plasticity. Connections between simple and complex cells are learned using a modified version of the Bienenstock, Cooper, and Munro (BCM) rule. Our results demonstrate that the learning rule can describe a diversity of complex cells, similar to those observed experimentally.

Implantable retinal stimulators activate surviving neurons to restore a sense of vision in people who have lost their photoreceptors through degenerative diseases. Complex spatial and temporal interactions occur in the retina during multi-electrode stimulation. Due to these complexities, most existing implants activate only a few electrodes at a time, limiting the repertoire of available stimulation patterns. Measuring the spatiotemporal interactions between electrodes and retinal cells, and incorporating them into a model may lead to improved stimulation algorithms that exploit the interactions. Here, we present a computational model that accurately predicts both the spatial and temporal nonlinear interactions of multi-electrode stimulation of rat retinal ganglion cells (RGCs). The model was verified using in vitro recordings of ON, OFF, and ON-OFF RGCs in response to subretinal multi-electrode stimulation with biphasic pulses at three stimulation frequencies (10, 20, 30 Hz). The model gives an estimate of each cell's spatiotemporal electrical receptive fields (ERFs); i.e., the pattern of stimulation leading to excitation or suppression in the neuron. All cells had excitatory ERFs and many also had suppressive sub-regions of their ERFs. We show that the nonlinearities in observed responses arise largely from activation of presynaptic interneurons. When synaptic transmission was blocked, the number of sub-regions of the ERF was reduced, usually to a single excitatory ERF. This suggests that direct cell activation can be modeled accurately by a one-dimensional model with linear interactions between electrodes, whereas indirect stimulation due to summated presynaptic responses is nonlinear.

The expansion of frontiers in neural engineering is dependent on the ability to track, detect and predict dynamics in neural tissue. Recent innovations to elucidate information from electrical recordings of brain dynamics, such as epileptic seizure prediction, have involved switching to an active probing paradigm using electrically evoked recordings rather than traditional passive measurements. This paper positions the advantage of probing in terms of information extraction, by using a coupled oscillator Kuramoto model to represent brain dynamics. While active probing performs better at observing underlying system synchrony in Kuramoto networks, especially in non-Gaussian measurement environments, the benefits diminish with increasing relative size of electrode spatial resolution compared to synchrony area. This suggests probing will be useful for improved characterization of synchrony for suitably dense electrode recordings.

Pitch perception is important for understanding speech prosody, music perception, recognizing tones in tonal languages, and perceiving speech in noisy environments. The two principal pitch perception theories consider the place of maximum neural excitation along the auditory nerve and the temporal pattern of the auditory neurons' action potentials (spikes) as pitch cues. This paper describes a biophysical mechanism by which fine-structure temporal information can be extracted from the spikes generated at the auditory periphery. Deriving meaningful pitch-related information from spike times requires neural structures specialized in capturing synchronous or correlated activity from amongst neural events. The emergence of such pitch-processing neural mechanisms is described through a computational model of auditory processing. Simulation results show that a correlation-based, unsupervised, spike-based form of Hebbian learning can explain the development of neural structures required for recognizing the pitch of simple and complex tones, with or without the fundamental frequency. The temporal code is robust to variations in the spectral shape of the signal and thus can explain the phenomenon of pitch constancy.

High-frequency oscillations (above 30 Hz) have been observed in sensory and higher-order brain areas, and are believed to constitute a general hallmark of functional neuronal activation. Fast inhibition in interneuronal networks has been suggested as a general mechanism for the generation of high-frequency oscillations. Certain classes of interneurons exhibit subthreshold oscillations, but the effect of this intrinsic neuronal property on the population rhythm is not completely understood. We study the influence of intrinsic damped subthreshold oscillations in the emergence of collective high-frequency oscillations, and elucidate the dynamical mechanisms that underlie this phenomenon. We simulate neuronal networks composed of either Integrate-and-Fire (IF) or Generalized Integrate-and-Fire (GIF) neurons. The IF model displays purely passive subthreshold dynamics, while the GIF model exhibits subthreshold damped oscillations. Individual neurons receive inhibitory synaptic currents mediated by spiking activity in their neighbors as well as noisy synaptic bombardment, and fire irregularly at a lower rate than population frequency. We identify three factors that affect the influence of single-neuron properties on synchronization mediated by inhibition: i) the firing rate response to the noisy background input, ii) the membrane potential distribution, and iii) the shape of Inhibitory Post-Synaptic Potentials (IPSPs). For hyperpolarizing inhibition, the GIF IPSP profile (factor iii)) exhibits post-inhibitory rebound, which induces a coherent spike-mediated depolarization across cells that greatly facilitates synchronous oscillations. This effect dominates the network dynamics, hence GIF networks display stronger oscillations than IF networks. However, the restorative current in the GIF neuron lowers firing rates and narrows the membrane potential distribution (factors i) and ii), respectively), which tend to decrease synchrony. If inhibition is shunting instead of hyperpolarizing, post-inhibitory rebound is not elicited and factors i) and ii) dominate, yielding lower synchrony in GIF networks than in IF networks.

Learning rules, such as spike-timing-dependent plasticity (STDP), change the structure of networks of neurons based on the firing activity. A network level understanding of these mechanisms can help infer how the brain learns patterns and processes information. Previous studies have shown that STDP selectively potentiates feed-forward connections that have specific axonal delays, and that this underlies behavioral functions such as sound localization in the auditory brainstem of the barn owl. In this study, we investigate how STDP leads to the selective potentiation of recurrent connections with different axonal and dendritic delays during oscillatory activity. We develop analytical models of learning with additive STDP in recurrent networks driven by oscillatory inputs, and support the results using simulations with leaky integrate-and-fire neurons. Our results show selective potentiation of connections with specific axonal delays, which depended on the input frequency. In addition, we demonstrate how this can lead to a network becoming selective in the amplitude of its oscillatory response to this frequency. We extend this model of axonal delay selection within a single recurrent network in two ways. First, we show the selective potentiation of connections with a range of both axonal and dendritic delays. Second, we show axonal delay selection between multiple groups receiving out-of-phase, oscillatory inputs. We discuss the application of these models to the formation and activation of neuronal ensembles or cell assemblies in the cortex, and also to missing fundamental pitch perception in the auditory brainstem.

Periodic neuronal activity has been observed in various areas of the brain, from lower sensory to higher cortical levels. Specific frequency components contained in this periodic activity can be identified by a neuronal circuit that behaves as a bandpass filter with given preferred frequency, or best modulation frequency (BMF). For BMFs typically ranging from 10 to 200 Hz, a plausible and minimal configuration consists of a single neuron with adjusted excitatory and inhibitory synaptic connections. The emergence, however, of such a neuronal circuitry is still unclear. In this letter, we demonstrate how spike-timing-dependent plasticity (STDP) can give rise to frequency-dependent learning, thus leading to an input selectivity that enables frequency identification. We use an in-depth mathematical analysis of the learning dynamics in a population of plastic inhibitory connections. These provide inhomogeneous postsynaptic responses that depend on their dendritic location. We find that synaptic delays play a crucial role in organizing the weight specialization induced by STDP. Under suitable conditions on the synaptic delays and postsynaptic potentials (PSPs), the BMF of a neuron after learning can match the training frequency. In particular, proximal (distal) synapses with shorter (longer) dendritic delay and somatically measured PSP time constants respond better to higher (lower) frequencies. As a result, the neuron will respond maximally to any stimulating frequency (in a given range) with which it has been trained in an unsupervised manner. The model predicts that synapses responding to a given BMF form clusters on dendritic branches.

Spike-timing-dependent plasticity (STDP) has been observed in many brain areas such as sensory cortices, where it is hypothesized to structure synaptic connections between neurons. Previous studies have demonstrated how STDP can capture spiking information at short timescales using specific input configurations, such as coincident spiking, spike patterns and oscillatory spike trains. However, the corresponding computation in the case of arbitrary input signals is still unclear. This paper provides an overarching picture of the algorithm inherent to STDP, tying together many previous results for commonly used models of pairwise STDP. For a single neuron with plastic excitatory synapses, we show how STDP performs a spectral analysis on the temporal cross-correlograms between its afferent spike trains. The postsynaptic responses and STDP learning window determine kernel functions that specify how the neuron "sees" the input correlations. We thus denote this unsupervised learning scheme as 'kernel spectral component analysis' (kSCA). In particular, the whole input correlation structure must be considered since all plastic synapses compete with each other. We find that kSCA is enhanced when weight-dependent STDP induces gradual synaptic competition. For a spiking neuron with a "linear" response and pairwise STDP alone, we find that kSCA resembles principal component analysis (PCA). However, plain STDP does not isolate correlation sources in general, e.g., when they are mixed among the input spike trains. In other words, it does not perform independent component analysis (ICA). Tuning the neuron to a single correlation source can be achieved when STDP is paired with a homeostatic mechanism that reinforces the competition between synaptic inputs. Our results suggest that neuronal networks equipped with STDP can process signals encoded in the transient spiking activity at the timescales of tens of milliseconds for usual STDP.

A closed-loop system for the automated detection and control of epileptic seizures was created and tested in three Genetic Absence Epilepsy Rats from Strasbourg (GAERS) rats. In this preliminary study, a set of four EEG features were used to detect seizures and three different electrical stimulation strategies (standard (130 Hz), very high (500 Hz) and ultra high (1000 Hz)) were delivered to terminate seizures. Seizure durations were significantly shorter with all three stimulation strategies when compared to non-stimulated (control) seizures. We used mean seizure duration of epileptiform discharges persisting beyond the end of electrical stimulation as a measure of stimulus efficacy. When compared to the duration of seizures stimulated in the standard approach (7.0 s ± 10.1), both very high and ultra high frequency stimulation strategies were more effective at shortening seizure durations (1.3 ± 2.2 s and 3.5 ± 6.4 s respectively). Further studies are warranted to further understand the mechanisms by which this therapeutic effect may be conveyed, and which of the novel aspects of the very high and ultra high frequency stimulation strategies may have contributed to the improvement in seizure abatement performance when compared to standard electrical stimulation approaches.

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.

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.

Cochlear implants, also known as bionic ears, are surgically implanted biomedical devices that can provide hearing to some deaf people by direct electrical stimulation of the auditory nerve. A crucial question for the design of future cochlear implants is that of how many electrodes might achieve optimal hearing performance in patients. It is efficient to avoid using more electrodes if this does not provide a performance improvement. Whether an improvement can be gained by inclusion of more electrodes depends crucially on physical properties such as distance of the electrode array from the auditory nerve and current spread. The response of individual fibers in the auditory nerve to electrical stimulation is stochastic, and it is proposed that the interface between an array of electrodes and the auditory nerve can be thought of as a communication channel in which only uncoded transmission can be used. A discrete memoryless channel model for this interface is defined and used as the basis for obtaining numerical estimates of the optimal number of electrodes in the array as a function of array-to-nerve distance. While the only true indicator of improved hearing through cochlear implants is via empirical audiological measurements, the discrete memoryless channel model allows maximization of mutual information as a proxy measure, under the hypothesis that there exists a monotonic relationship between mutual information and perceptibility.

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.

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.

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.

The observed variability in the spike rate of cortical neurons has been hypothesized to result from a balance in the excitatory and inhibitory synaptic inputs that the neurons receive. The coefficient of variation of the spike rate is calculated in the limit of a large number of inputs using the integrated-input technique, which is extended here to include the effect of reversal potentials. The output spike rate is found to increase monotonically over two orders of magnitude, thereby solving the dynamic range (or gain control) problem. The coefficient of variation is approximately 1.0 for low input rates and increases to around 1.6 at high input rates, well within the range observed in the response of cortical neurons.

The relationship between the timing of the synaptic inputs and the output spikes of leaky integrated and firing neurons with noisy periodic synaptic input is addressed using the recently developed integrated-input technique. 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 dependence of the synchronization index of the neural response upon the number and amplitude of synaptic inputs, the membrane time constant, the average rate of inputs and their frequency of modulation is examined.

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.