Different types of noise in leaky integrate-and-fire model of neuronal dynamics with discrete periodical input

Different variants of stochastic leaky integrate-and-fire model for the membrane depolarisation of neurons are investigated. The model is driven by a constant input and equidistant pulses of fixed amplitude. These two types of signal are considered under the influence of three types of noise: white noise, jitter on interpulse distance, and noise in the amplitude of pulses. The results of computational experiments demonstrate the enhancement of the signal by noise in subthreshold regime and deterioration of the signal if it is sufficiently strong to carry the information in absence of noise. Our study holds mainly to central neurons that process discrete pulses although an application in sensory system is also available.     

Optimum signal in a diffusion leaky integrate-and-fire neuronal model

An optimum signal in the Ornstein-Uhlenbeck neuronal model is determined on the basis of interspike interval data. Two criteria are proposed for this purpose. The first, the classical one, is based on searching for maxima of the slope of the frequency transfer function. The second one uses maximum of the Fisher information, which is, under certain conditions, the inverse variance of the best possible estimator. The Fisher information is further normalized with respect to the time required to make the observation on which the signal estimation is performed. Three variants of the model are investigated. Beside the basic one, we use the version obtained by inclusion of the refractory period. Finally, we investigate such a version of the model in which signal and the input parameter of the model are in a nonlinear relationship. The results show that despite qualitative similarity between the criteria, there is substantial quantitative difference. As a common feature, we found that in the Ornstein-Uhlenbeck model with increasing noise the optimum signal decreases and the coding range gets broader.     

The parameters of the stochastic leaky integrate-and-fire neuronal model

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data. Action Editor: Nicolas Brunel     

Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant tau is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. The computational methods described here are a efficient alternative to other well known estimation techniques for leaky integrate-and-fire models. Moreover, it could serve as a template for performing parameter inference on more complex integrate-and-fire neuronal models.     

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).     

Balance between four types of synaptic input for the integrate-and-fire model

We consider the integrate-and-fire model with AMPA, NMDA, GABA(A)and GABA(B)synaptic inputs, with model parameters based upon experimental data. An analytical approach is presented to determine when a post-synaptic balance between excitation and inhibition can be achieved. Secondly, we compare the model behaviour subject to these four types of input, with its behaviour subject to conventional point process inputs. We conclude that point processes are not a good approximation, even away from exact presynaptic balance. Thirdly, numerical simulations are presented which demonstrate that we can treat NMDA and GABA(B)as DC currents. Finally, we conclude that a balanced input is plausible neither pre-synaptically nor post-synaptically for the model and parameters we employed.     

Interrelated modification of excitatory and inhibitory synaptic connections in the olivary-cerebellar neuronal network

The model of simultaneous interrelated modification in the efficacy of synaptic inputs to different neurons of the olivary-cerebellar network is developed. The model is based on the following features of the network: simultaneous activation of the input layer (granule) cells and the output layer (deep cerebellar nuclei) cells by mossy fibers; simultaneous activation of Purkinje cells and cerebellar cells of the input and output layers by climbing fibers and their collaterals; the existence of local feedback excitatory, inhibitory, and disinhibitory circuits. The rise (decrease) of posttetanic Ca2+ concentration in reference to the level produced by previous stimulation causes the decrease (increase) in cGMP-dependent protein kinase G activity, and increase (decrease) inprotein phosphatase 1 activity. Subsequent dephosphorylation (phosphorylation) of ionotropic receptors results in simultaneous LTD (LTP) of the excitatory input together with the LTP (LTD) of the inhibitory input to the same neuron. The character of interrelated modifications of synapses at different cerebellar levels strongly depends on the olivary cell activity. In the presence (absence) of the signal from the inferior olive LTD (LTP) of the output cerebellar signal can be induced.     

Maternal loss of Ube3a produces an excitatory/inhibitory imbalance through neuron type-specific synaptic defects

Angelman syndrome (AS) is a neurodevelopmental disorder caused by loss of the maternally inherited allele of UBE3A. AS model mice, which carry a maternal Ube3a null mutation (Ube3a(m-/p+)), recapitulate major features of AS in humans, including enhanced seizure susceptibility. Excitatory neurotransmission onto neocortical pyramidal neurons is diminished in Ube3a(m-/p+) mice, seemingly at odds with enhanced seizure susceptibility. We show here that inhibitory drive onto neocortical pyramidal neurons is more severely decreased in Ube3a(m-/p+) mice. This inhibitory deficit follows the loss of excitatory inputs and appears to arise from defective presynaptic vesicle cycling in multiple interneuron populations. In contrast, excitatory and inhibitory synaptic inputs onto inhibitory interneurons are largely normal. Our results indicate that there are neuron type-specific synaptic deficits in Ube3a(m-/p+) mice despite the presence of Ube3a in all neurons. These deficits result in excitatory/inhibitory imbalance at cellular and circuit levels and may contribute to seizure susceptibility in AS.     

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave.     

A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models

Parameters in diffusion neuronal models are divided into two groups; intrinsic and input parameters. Intrinsic parameters are related to the properties of the neuronal membrane and are assumed to be known throughout the paper. Input parameters characterize processes generated outside the neuron and methods for their estimation are reviewed here. Two examples of the diffusion neuronal model, which are based on the integrate-and-fire concept, are investigated--the Ornstein--Uhlenbeck model as the most common one and the Feller model as an illustration of state-dependent behavior in modeling the neuronal input. Two types of experimental data are assumed-intracellular describing the membrane trajectories and extracellular resulting in knowledge of the interspike intervals. The literature on estimation from the trajectories of the diffusion process is extensive and thus the stress in this review is set on the inference made from the interspike intervals.     

Relation between the excitatory synaptic currents and clamped somatic potential in the model of neuron with nonlinear dendrites

The influence of the clamped somatic potential on the excitatory synaptic current (EPSC) was studied in the model of the dendrite with N-shaped instantaneous stationary current--voltage curve. Proximal EPSC diminish and become narrower with decreasing hyperpolarization or modest depolarization, distal EPSC increase and become wider, intermediately distant EPSC change insignificantly. Under increasing depolarization all the EPSC become significantly wider and larger. EPSC facilitate stable depolarization of the dendrites. When the dendrite is stable depolarized EPSC becomes very small and narrow, but it becomes larger and wider as the soma is hyperpolarized. EPSC becomes especially large and wide when the soma is hyperpolarized just to terminate the stable depolarization of the dendrite branch where the active synapses are located. The model explains certain phenomena which are difficult to understand by the theory of ohmic dendrites. New phenomena are predicted.     

Firing patterns in a conductance-based neuron model: bifurcation, phase diagram, and chaos

Responding to various stimuli, some neurons either remain resting or can fire several distinct patterns of action potentials, such as spiking, bursting, subthreshold oscillations, and chaotic firing. In particular, Wilson’s conductance-based neocortical neuron model, derived from the Hodgkin–Huxley model, is explored to understand underlying mechanisms of the firing patterns. Phase diagrams describing boundaries between the domains of different firing patterns are obtained via extensive numerical computations. The boundaries are further studied by standard instability analyses, which demonstrates that the chaotic neural firing could develop via period-doubling and/or period- adding cascades. Sequences of the firing patterns often observed in many neural experiments are also discussed in the phase diagram framework developed. Our results lay the groundwork for wider use of the model, especially for incorporating it into neural field modeling of the brain.     

Differences in synaptic output between excitatory and inhibitory motoneurons in a crayfish muscle

A pair of antagonistic motoneurons, one excitatory and one inhibitory, innervates the distal accessory flexor muscle in the walking limb of the crayfish Procambarus clarkii. The number and size of synapses formed by these two axons on the muscle fibers (neuromuscular synapses) and on each other (axo-axonal synapses) were estimated using thin-section electron microscopy. Although profiles of nerve terminals of the two axons occur in roughly equal proportions, the frequency of occurrence of neuromuscular synapses differed markedly: 73% were excitatory and 27% were inhibitory. However, inhibitory synapses were 4–5 times larger than excitatory ones, and consequently, the total contact areas devoted to neuromuscular synapses were similar for both axons. Axo-axonal synapses were predominantly from the inhibitory axon to the excitatory axon (86%), and a few were from the excitatory axon to the inhibitory axon (14%). The role of the inhibitory axo-axonal synapse is presynaptic inhibition, but that of the excitatory axo-axonal synapse is not known. The differences in size of neuromuscular synapses between the two axons may reflect intrinsic determinants of the neuron, while the similarity in total synaptic area may reflect retrograde influences from the muscle for regulating synapse number.     

Method of evaluating the synaptic input to a single mesencephalic neuron]

The mathematical model of the spike activity of a neuron with synaptic input from many other neurons [1], describes adequately the firing of 5 from 7 neurons in the tegmentum of mesencephalic cat and changes of their activity evoked by glutamate iontophoresis. For these 5 neurons the estimates of the PSPs' average frequency of the threshold depolarization and of the constant decay of the EPSP were received. For different neurons the values of these parameters are 4--100 KHz, 100--800 average unitary EPSPs and 4--30 msec correspondingly. The stationary value of the average membrane potential (SVAMP) in all 5 neurons was removed significantly from the resting potential toward the threshold potential. SWAMP could be changed by the glutamate iontophoresis in such a degree to overlap the threshold potential.