Depressing synapse as a detector of frequency change

In this article we discuss the short-term synaptic depression using a mathematical model. We derive the model of synaptic depression caused by the depletion of synaptic vesicles for the case of infinitely short stimulation time and show that the analytical formulas for the postsynaptic potential (PSP) and kinetic functions take simple closed form. A solution in this form allows an analysis of the characteristics of depression as a function of the models parameters and the derivation of analytic formulas for measures of short time synaptic depression commonly used in experimental studies. Those formulas are used to validate the model by fitting it to two types of synapses described in the literature. Given the fitted parameters we discuss the behavior of the synapse in situations involving frequency change. We also indicate a possible role of depressing synapses in information processing as not only a filter of high frequency input but as a detector of the return from high frequency stimulation to the stimulation within frequency band specific for a given synapse.     

A Calcium-Dependent Plasticity Rule for HCN Channels Maintains Activity Homeostasis and Stable Synaptic Learning

Theoretical and computational frameworks for synaptic plasticity and learning have a long and cherished history, with few parallels within the well-established literature for plasticity of voltage-gated ion channels. In this study, we derive rules for plasticity in the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, and assess the synergy between synaptic and HCN channel plasticity in establishing stability during synaptic learning. To do this, we employ a conductance-based model for the hippocampal pyramidal neuron, and incorporate synaptic plasticity through the well-established Bienenstock-Cooper-Munro (BCM)-like rule for synaptic plasticity, wherein the direction and strength of the plasticity is dependent on the concentration of calcium influx. Under this framework, we derive a rule for HCN channel plasticity to establish homeostasis in synaptically-driven firing rate, and incorporate such plasticity into our model. In demonstrating that this rule for HCN channel plasticity helps maintain firing rate homeostasis after bidirectional synaptic plasticity, we observe a linear relationship between synaptic plasticity and HCN channel plasticity for maintaining firing rate homeostasis. Motivated by this linear relationship, we derive a calcium-dependent rule for HCN-channel plasticity, and demonstrate that firing rate homeostasis is maintained in the face of synaptic plasticity when moderate and high levels of cytosolic calcium influx induced depression and potentiation of the HCN-channel conductance, respectively. Additionally, we show that such synergy between synaptic and HCN-channel plasticity enhances the stability of synaptic learning through metaplasticity in the BCM-like synaptic plasticity profile. Finally, we demonstrate that the synergistic interaction between synaptic and HCN-channel plasticity preserves robustness of information transfer across the neuron under a rate-coding schema. Our results establish specific physiological roles for experimentally observed plasticity in HCN channels accompanying synaptic plasticity in hippocampal neurons, and uncover potential links between HCN-channel plasticity and calcium influx, dynamic gain control and stable synaptic learning.     

Neural field model of binocular rivalry waves

We present a neural field model of binocular rivalry waves in visual cortex. For each eye we consider a one-dimensional network of neurons that respond maximally to a particular feature of the corresponding image such as the orientation of a grating stimulus. Recurrent connections within each one-dimensional network are assumed to be excitatory, whereas connections between the two networks are inhibitory (cross-inhibition). Slow adaptation is incorporated into the model by taking the network connections to exhibit synaptic depression. We derive an analytical expression for the speed of a binocular rivalry wave as a function of various neurophysiological parameters, and show how properties of the wave are consistent with the wave-like propagation of perceptual dominance observed in recent psychophysical experiments. In addition to providing an analytical framework for studying binocular rivalry waves, we show how neural field methods provide insights into the mechanisms underlying the generation of the waves. In particular, we highlight the important role of slow adaptation in providing a “symmetry breaking mechanism” that allows waves to propagate.     

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye.     

The Activity Phase of Postsynaptic Neurons in a Simplified Rhythmic Network

Many inhibitory rhythmic networks produce activity in a range of frequencies. The relative phase of activity between neurons in these networks is often a determinant of the network output. This relative phase is determined by the interaction between synaptic inputs to the neurons and their intrinsic properties. We show, in a simplified network consisting of an oscillator inhibiting a follower neuron, how the interaction between synaptic depression and a transient potassium current in the follower neuron determines the activity phase of this neuron. We derive a mathematical expression to determine at what phase of the oscillation the follower neuron becomes active. This expression can be used to understand which parameters determine the phase of activity of the follower as the frequency of the oscillator is changed. We show that in the presence of synaptic depression, there can be three distinct frequency intervals, in which the phase of the follower neuron is determined by different sets of parameters. Alternatively, when the synapse is not depressing, only one set of parameters determines the phase of activity at all frequencies.     

A Memristor SPICE Model Accounting for Synaptic Activity Dependence

In this work, we propose a new memristor SPICE model that accounts for the typical synaptic characteristics that have been previously demonstrated with practical memristive devices. We show that this model could account for both volatile and non-volatile memristance changes under distinct stimuli. We then demonstrate that our model is capable of supporting typical STDP with simple non-overlapping digital pulse pairs. Finally, we investigate the capability of our model to simulate the activity dependence dynamics of synaptic modification and present simulated results that are in excellent agreement with biological results.     

Kinetics of synaptic depression and vesicle recycling after tetanic stimulation of frog motor nerve terminals.

We measured the time courses of two key components of the synaptic vesicle cycle during recovery from synaptic depression under different conditions, and used this and other information to create a kinetic model of the vesicle cycle. End plate potential (EPP) amplitudes were used to follow recovery from synaptic depression after different amounts of tetanic stimulation. This provided an estimate of the time course of vesicle mobilization from the reserve pool to the docked (readily releasable) pool. In addition, FM1-43 was used to measure the rate of membrane retrieval after tetanic stimulation, and the amount of membrane transferred to the surface membrane. This provided a measure of the rate of refilling of the reserve pool with recycled vesicles. The time courses of both synaptic depression and endocytosis were slowed by prolonged tetanic stimulation. This behavior could be fitted by a simple model, assuming a first-order kinetics for both vesicle endocytosis and mobilization. The results show that a nearly 20-fold decrease in the rate constant of endocytosis greatly delays refilling of the depleted reserve pool. However, to fully account for the slower recovery of depression, a decrease in the rate constant of vesicle mobilization from the reserve pool of about sixfold is also required.     

Novel Bursting Patterns Emerging from Model Inhibitory Networks with Synaptic Depression

Studies show that short-term synaptic plasticity plays important roles in neural coding and the normal operation of the synapse. Basket cells in the hippocampus demonstrate this plasticity in the form of synaptic depression, and recent in vivo work indicates that basket cell activities contribute significantly to hippocampal output associated with different behavioural states. Thus it is essential to understand the generation and synchronization of patterns produced by basket cell networks with depression. We study two-cell model inhibitory networks with depression and obtain alternating bursting patterns and synchronous activity occurring between bursts. We describe mechanisms for how these patterns emerge by performing several simulations in the plane of different depression time constants, tauD. Such patterns might contribute significantly to various population activities observed in the hippocampus.     

Efficient computation in the IM model

In this paper we analyze the isolation-with-migration model in a continuous-time Markov chain framework, and derive analytical expressions for the probability densities of gene tree topologies with an arbitrary number of lineages. We combine these densities with both nucleotide-substitution and infinite sites mutation models and derive probabilities for use in maximum likelihood estimation. We demonstrate how to apply lumpability of continuous-time Markov chains to achieve a significant reduction in the size of the state-space under consideration. We use matrix exponentiation and spectral decomposition to derive explicit expressions for the case of two diploid individuals in two populations, when the data is given as alignment columns. We implement these expressions in order to carry out a maximum likelihood analysis and provide a simulation study to examine the performance of our method in terms of our ability to recover true parameters. Finally, we show how the performance depends on the parameters in the model.     

Resource allocation for epidemic control over short time horizons

We present a model for allocation of epidemic control resources among a set of interventions. We assume that the epidemic is modeled by a general compartmental epidemic model, and that interventions change one or more of the parameters that describe the epidemic. Associated with each intervention is a 'production function' that relates the amount invested in the intervention to values of parameters in the epidemic model. The goal is to maximize quality-adjusted life years gained or the number of new infections averted over a fixed time horizon, subject to a budget constraint. Unlike previous models, our model allows for interacting populations and non-linear interacting production functions and does not require a long time horizon. We show that an analytical solution to the model may be difficult or impossible to derive, even for simple cases. Therefore, we derive a method of approximating the objective functions. We use the approximations to gain insight into the optimal resource allocation for three problem instances. We also develop heuristics for solving the general resource allocation problem. We present results of numerical studies using our approximations and heuristics. Finally, we discuss implications and applications of this work.     

Extracellular Ca2+ depletion contributes to fast activity-dependent modulation of synaptic transmission in the brain

Synaptic activation is associated with rapid changes in intracellular Ca(2+), while the extracellular Ca(2+) level is generally assumed to be constant. Here, using a novel optical method to measure changes in extracellular Ca(2+) at high spatial and temporal resolution, we find that brief trains of synaptic transmission in hippocampal area CA1 induce transient depletion of extracellular Ca(2+). We show that this depletion, which depends on postsynaptic NMDA receptor activation, decreases the Ca(2+) available to enter individual presynaptic boutons of CA3 pyramidal cells. This in turn reduces the probability of consecutive synaptic releases at CA3-CA1 synapses and therefore contributes to short-term paired-pulse depression of minimal responses. This activity-dependent depletion of extracellular Ca(2+) represents a novel form of fast retrograde synaptic signaling that can modulate glutamatergic information transfer in the brain.     

Democratization in a passive dendritic tree: an analytical investigation

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as "dendritic democracy". Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green's representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.     

Short term synaptic depression imposes a frequency dependent filter on synaptic information transfer

Depletion of synaptic neurotransmitter vesicles induces a form of short term depression in synapses throughout the nervous system. This plasticity affects how synapses filter presynaptic spike trains. The filtering properties of short term depression are often studied using a deterministic synapse model that predicts the mean synaptic response to a presynaptic spike train, but ignores variability introduced by the probabilistic nature of vesicle release and stochasticity in synaptic recovery time. We show that this additional variability has important consequences for the synaptic filtering of presynaptic information. In particular, a synapse model with stochastic vesicle dynamics suppresses information encoded at lower frequencies more than information encoded at higher frequencies, while a model that ignores this stochasticity transfers information encoded at any frequency equally well. This distinction between the two models persists even when large numbers of synaptic contacts are considered. Our study provides strong evidence that the stochastic nature neurotransmitter vesicle dynamics must be considered when analyzing the information flow across a synapse.     

The effects of synaptic noise on measurements of evoked excitatory postsynaptic response amplitudes.

Spontaneously occurring synaptic events (synaptic noise) recorded intracellularly are usually assumed to be independent of evoked postsynaptic responses and to contaminate measures of postsynaptic response amplitude in a roughly Gaussian manner. Here we derive analytically the expected noise distribution for excitatory synaptic noise and investigate its effects on amplitude histograms. We propose that some fraction of this excitatory noise is initiated at the same release sites that contribute to the evoked synaptic event and develop an analytical model of the interaction between this fraction of the noise and the evoked postsynaptic response amplitude. Recording intracellularly with sharp microelectrodes in the in vitro hippocampal slice preparation, we find that excitatory synaptic noise accounts for up to 70% of the intracellular recording noise, when inhibition is blocked pharmacologically. Up to 20% of this noise shows a significant correlation with the evoked event amplitude, and the behavior of this component of the noise is consistent with a model which assumes that each release site experiences a refractory period of approximately 60 ms after release. In contrast with classical models of quantal variance, our models predict that excitatory synaptic noise can cause the apparent variance of successive peaks in an excitatory synaptic amplitude histogram to decrease from left to right, and in some cases to be less than the variance of the measured noise.     

Synapse geometry and receptor dynamics modulate synaptic strength

Synaptic transmission relies on several processes, such as the location of a released vesicle, the number and type of receptors, trafficking between the postsynaptic density (PSD) and extrasynaptic compartment, as well as the synapse organization. To study the impact of these parameters on excitatory synaptic transmission, we present a computational model for the fast AMPA-receptor mediated synaptic current. We show that in addition to the vesicular release probability, due to variations in their release locations and the AMPAR distribution, the postsynaptic current amplitude has a large variance, making a synapse an intrinsic unreliable device. We use our model to examine our experimental data recorded from CA1 mice hippocampal slices to study the differences between mEPSC and evoked EPSC variance. The synaptic current but not the coefficient of variation is maximal when the active zone where vesicles are released is apposed to the PSD. Moreover, we find that for certain type of synapses, receptor trafficking can affect the magnitude of synaptic depression. Finally, we demonstrate that perisynaptic microdomains located outside the PSD impacts synaptic transmission by regulating the number of desensitized receptors and their trafficking to the PSD. We conclude that geometrical modifications, reorganization of the PSD or perisynaptic microdomains modulate synaptic strength, as the mechanisms underlying long-term plasticity.     

Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

 We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation. Received: 27 October 1998     

A simplified physical model of pressure wave dynamics and acoustic wave generation induced by laser absorption in the retina

Shock waves have been proposed in the literature as a mechanism for retinal damage induced by ultra-short laser pulses. For a spherical absorber, we derive a set of linear equations describing the propagation of pressure waves. We show that the formation of shock fronts is due to the form of the absorber rather than the inclusion of nonlinear terms in the equations. The analytical technique used avoids the need for a Laplace transform approach and is easily applied to other absorber profiles. Our analysis suggests that the ’soft’ nature of the membrane surrounding retinal melanosomes precludes shock waves as a mechanism for the retinal damage induced by ultra-short pulse lasers. The quantitative estimates of the pressure gradients induced by laser absorption which are made possible by this work, together with detailed meso-scale or molecular modelling, will allow alternative damage mechanisms to be identified.     

Inclusive fitness for traits affecting metapopulation demography

Defining computable analytical measures of the effects of selection in populations with demographic and environmental stochasticity is a long-standing problem. We derive an analytical measure which takes in account all consequences of the discrete nature of deme size. Expressions of this measure are detailed for infinite island models of population structure. As an illustration we consider the evolution of dispersal in populations made of small demes with environmental and demographic stochasticity. We confirm some results obtained from the analysis of models based on deterministic approximations. In particular, when there is an Allee effect, we show that evolution of the dispersal rate may lead the metapopulation to extinction. Thus, selection on the dispersal rate could restrict the distribution of species subject to Allee effects. This selection-driven extinction is prevented by kin selection when the environmental extinction rate is small.