Abstract:
Local cortical circuits appear highly non-random, but the underlying connectivity rule remains elusive. Here, we analyze experimental data observed in layer 5 of rat neocortex and suggest a model for connectivity from which emerge essential observed non-random features of both wiring and weighting. These features include lognormal distributions of synaptic connection strength, anatomical clustering, and strong correlations between clustering and connection strength. Our model predicts that cortical microcircuits contain large groups of densely connected neurons which we call clusters. We show that such a cluster contains about one fifth of all excitatory neurons of a circuit which are very densely connected with stronger than average synapses. We demonstrate that such clustering plays an important role in the network dynamics, namely, it creates bistable neural spiking in small cortical circuits. Furthermore, introducing local clustering in large-scale networks leads to the emergence of various patterns of persistent local activity in an ongoing network activity. Thus, our results may bridge a gap between anatomical structure and persistent activity observed during working memory and other cognitive processes.

Abstract:
We consider a new class of non Markovian processes with a countable number of interacting components, both in discrete and continuous time. Each component is represented by a point process indicating if it has a spike or not at a given time. The system evolves as follows. For each component, the rate (in continuous time) or the probability (in discrete time) of having a spike depends on the entire time evolution of the system since the last spike time of the component. In discrete time this class of systems extends in a non trivial way both Spitzer's interacting particle systems, which are Markovian, and Rissanen's stochastic chains with memory of variable length which have finite state space. In continuous time they can be seen as a kind of Rissanen's variable length memory version of the class of self-exciting point processes which are also called "Hawkes processes", however with infinitely many components. These features make this class a good candidate to describe the time evolution of networks of spiking neurons. In this article we present a critical reader's guide to recent papers dealing with this class of models, both in discrete and in continuous time. We briefly sketch results concerning perfect simulation and existence issues, de-correlation between successive interspike intervals, the longtime behavior of finite non-excited systems and propagation of chaos in mean field systems.

Abstract:
We study associative memory neural networks of the Hodgkin-Huxley type of spiking neurons in which multiple periodic spatio-temporal patterns of spike timing are memorized as limit-cycle-type attractors. In encoding the spatio-temporal patterns, we assume the spike-timing-dependent synaptic plasticity with the asymmetric time window. Analysis for periodic solution of retrieval state reveals that if the area of the negative part of the time window is equivalent to the positive part, then crosstalk among encoded patterns vanishes. Phase transition due to the loss of the stability of periodic solution is observed when we assume fast alpha-function for direct interaction among neurons. In order to evaluate the critical point of this phase transition, we employ Floquet theory in which the stability problem of the infinite number of spiking neurons interacting with alpha-function is reduced into the eigenvalue problem with the finite size of matrix. Numerical integration of the single-body dynamics yields the explicit value of the matrix, which enables us to determine the critical point of the phase transition with a high degree of precision.

Abstract:
Precise temporal synchrony of spike firing has been postulated as an important neuronal mechanism for signal integration and the induction of plasticity in neocortex. As prefrontal cortex plays an important role in organizing memory and executive functions, the convergence of multiple visual pathways onto PFC predicts that neurons should preferentially synchronize their spiking when stimulus information is processed. Furthermore, synchronous spike firing should intensify if memory processes require the induction of neuronal plasticity, even if this is only for short-term. Here we show with multiple simultaneously recorded units in ventral prefrontal cortex that neurons participate in 3 ms precise synchronous discharges distributed across multiple sites separated by at least 500 μm. The frequency of synchronous firing is modulated by behavioral performance and is specific for the memorized visual stimuli. In particular, during the memory period in which activity is not stimulus driven, larger groups of up to seven sites exhibit performance dependent modulation of their spike synchronization.

Abstract:
Humans and other animals behave as if we perform fast Bayesian inference underlying decisions and movement control given uncertain sense data. Here we show that a biophysically realistic model of the subthreshold membrane potential of a single neuron can exactly compute the numerator in Bayes rule for inferring the Poisson parameter of a sensory spike train. A simple network of spiking neurons can construct and represent the Bayesian posterior density of a parameter of an external cause that affects the Poisson parameter, accurately and in real time.

Abstract:
Although models based on independent component analysis (ICA) have been successful in explaining various properties of sensory coding in the cortex, it remains unclear how networks of spiking neurons using realistic plasticity rules can realize such computation. Here, we propose a biologically plausible mechanism for ICA-like learning with spiking neurons. Our model combines spike-timing dependent plasticity and synaptic scaling with an intrinsic plasticity rule that regulates neuronal excitability to maximize information transmission. We show that a stochastically spiking neuron learns one independent component for inputs encoded either as rates or using spike-spike correlations. Furthermore, different independent components can be recovered, when the activity of different neurons is decorrelated by adaptive lateral inhibition.

Abstract:
Recent experimental work has suggested that the neural firing rate can be interpreted as a fractional derivative, at least when signal variation induces neural adaptation. Here, we show that the actual neural spike-train itself can be considered as the fractional derivative, provided that the neural signal is approximated by a sum of power-law kernels. A simple standard thresholding spiking neuron suffices to carry out such an approximation, given a suitable refractory response. Empirically, we find that the online approximation of signals with a sum of power-law kernels is beneficial for encoding signals with slowly varying components, like long-memory self-similar signals. For such signals, the online power-law kernel approximation typically required less than half the number of spikes for similar SNR as compared to sums of similar but exponentially decaying kernels. As power-law kernels can be accurately approximated using sums or cascades of weighted exponentials, we demonstrate that the corresponding decoding of spike-trains by a receiving neuron allows for natural and transparent temporal signal filtering by tuning the weights of the decoding kernel.

Abstract:
Stress, pervasive in society, contributes to over half of all work place accidents a year and over time can contribute to a variety of psychiatric disorders including depression, schizophrenia, and post-traumatic stress disorder. Stress impairs higher cognitive processes, dependent on the prefrontal cortex (PFC) and that involve maintenance and integration of information over extended periods, including working memory and attention. Substantial evidence has demonstrated a relationship between patterns of PFC neuron spiking activity (action-potential discharge) and components of delayed-response tasks used to probe PFC-dependent cognitive function in rats and monkeys. During delay periods of these tasks, persistent spiking activity is posited to be essential for the maintenance of information for working memory and attention. However, the degree to which stress-induced impairment in PFC-dependent cognition involves changes in task-related spiking rates or the ability for PFC neurons to retain information over time remains unknown. In the current study, spiking activity was recorded from the medial PFC of rats performing a delayed-response task of working memory during acute noise stress (93 db). Spike history-predicted discharge (SHPD) for PFC neurons was quantified as a measure of the degree to which ongoing neuronal discharge can be predicted by past spiking activity and reflects the degree to which past information is retained by these neurons over time. We found that PFC neuron discharge is predicted by their past spiking patterns for nearly one second. Acute stress impaired SHPD, selectively during delay intervals of the task, and simultaneously impaired task performance. Despite the reduction in delay-related SHPD, stress increased delay-related spiking rates. These findings suggest that neural codes utilizing SHPD within PFC networks likely reflects an additional important neurophysiological mechanism for maintenance of past information over time. Stress-related impairment of this mechanism is posited to contribute to the cognition-impairing actions of stress.

Abstract:
Many central neurons, and in particular certain brainstem aminergic neurons exhibit spontaneous and fairly regular spiking with frequencies of order a few Hz. A large number of ion channel types contribute to such spiking so that accurate modeling of spike generation leads to the requirement of solving very large systems of differential equations, ordinary in the first instance. Since analysis of spiking behavior when many synaptic inputs are active adds further to the number of components, it is useful to have simplified mathematical models of spiking in such neurons so that, for example, stochastic features of inputs and output spike trains can be incorporated. In this article we investigate two simple two-component models which mimic features of spiking in serotonergic neurons of the dorsal raphe nucleus and noradrenergic neurons of the locus coeruleus. The first model is of the Fitzhugh-Nagumo type and the second is a reduced Hodgkin-Huxley model. For each model solutions are computed with two representative sets of parameters. Frequency versus input currents are found and reveal Hodgkin type 2 behavior. For the first model a bifurcation and phase plane analysis supports these findings. The spike trajectories in the second model are very similar to those in DRN SE pacemaker activity but there are more parameters than in the Fitzhugh-Nagumo type model. The article concludes with a brief review of previous modeling of these types of neurons and its relevance to studies of serotonergic involvement in spatial working memory and obsessive-compulsive disorder.

Abstract:
A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the underlying microscopic state of the spiking neurons. This has largely constrained the range of applicability of such macroscopic descriptions, particularly when trying to describe neuronal synchronization. Here we provide the derivation of a set of exact macroscopic equations for a network of spiking neurons. Our results reveal that the spike generation mechanism of individual neurons introduces an effective coupling between two biophysically relevant macroscopic quantities, the firing rate and the mean membrane potential, which together govern the evolution of the neuronal network. The resulting equations exactly describe all possible macroscopic dynamical states of the network, including states of synchronous spiking activity. Finally we show that the firing rate description is related, via a conformal map, with a low-dimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate our results will be an important tool in investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.