Local accumulation times in a diffusion-trapping model of receptor dynamics at proximal axodendritic synapses.

The lateral diffusion and trapping of neurotransmitter receptors within the postsynaptic membrane of a neuron play a key role in determining synaptic strength and plasticity. Trapping is mediated by the reversible binding of receptors to scaffolding proteins (slots) within a synapse. In this paper we introduce a method for analyzing the transient dynamics of proximal axodendritic synapses in a diffusion-trapping model of receptor trafficking. Given a population of spatially distributed synapses, each of which has a fixed number of slots, we calculate the rate of relaxation to the steady-state distribution of bound slots (synaptic weights) in terms of a set of local accumulation times. Assuming that the rates of exocytosis and endocytosis are sufficiently slow, we show that the steady-state synaptic weights are independent of each other (purely local). On the other hand, the local accumulation time of a given synapse depends on the number of slots and the spatial location of all the synapses, indicating a form of transient heterosynaptic plasticity. This suggests that local accumulation time measurements could provide useful information regarding the distribution of synaptic weights within a dendrite.

Target competition for resources under multiple search-and-capture events with stochastic resetting.

We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location x r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to x r , where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with N small interior targets. We show that slow resetting increases the total number of resources M tot across all targets provided that ∑ j = 1 N G ( x r , x j ) < 0 , where G is the Neumann Green's function and x j is the location of the j-th target. This implies that M tot can be optimized by varying r. We also show that the k-th target has a competitive advantage if ∑ j = 1 N G ( x r , x j ) > N G ( x r , x k ) .

Propagation of Extrinsic Fluctuations in Biochemical Birth-Death Processes.

Biochemical reactions are often subject to a complex fluctuating environment, which means that the corresponding reaction rates may themselves be time-varying and stochastic. If the environmental noise is common to a population of downstream processes, then the resulting rate fluctuations will induce statistical correlations between them. In this paper we investigate how such correlations depend on the form of environmental noise by considering a simple birth-death process with dynamical disorder in the birth rate. In particular, we derive expressions for the second-order statistics of two birth-death processes evolving in the same noisy environment. We find that these statistics not only depend on the second-order statistics of the environment, but the full generator of the process describing it, thus providing useful information about the environment. We illustrate our theory by considering applications to stochastic gene transcription and cell sensing.

Synaptic bistability due to nucleation and evaporation of receptor clusters.

We introduce a bistability mechanism for long-term synaptic plasticity based on switching between two metastable states that contain significantly different numbers of synaptic receptors. One state is characterized by a two-dimensional gas of mobile interacting receptors and is stabilized against clustering by a high nucleation barrier. The other state contains a receptor gas in equilibrium with a large cluster of immobile receptors, which is stabilized by the turnover rate of receptors into and out of the synapse. Transitions between the two states can be initiated by either an increase (potentiation) or a decrease (depotentiation) of the net receptor flux into the synapse. This changes the saturation level of the receptor gas and triggers nucleation or evaporation of receptor clusters.

Modeling the role of lateral membrane diffusion in AMPA receptor trafficking along a spiny dendrite.

AMPA receptor trafficking in dendritic spines is emerging as a major postsynaptic mechanism for the expression of plasticity at glutamatergic synapses. AMPA receptors within a spine are in a continuous state of flux, being exchanged with local intracellular pools via exo/endocytosis and with the surrounding dendrite via lateral membrane diffusion. This suggests that one cannot treat a single spine in isolation. Here we present a model of AMPA receptor trafficking between multiple dendritic spines distributed along the surface of a dendrite. Receptors undergo lateral diffusion within the dendritic membrane, with each spine acting as a spatially localized trap where receptors can bind to scaffolding proteins or be internalized through endocytosis. Exocytosis of receptors occurs either at the soma or at sites local to dendritic spines via constitutive recycling from intracellular pools. We derive a reaction-diffusion equation for receptor trafficking that takes into account these various processes. Solutions of this equation allow us to calculate the distribution of synaptic receptor numbers across the population of spines, and hence determine how lateral diffusion contributes to the strength of a synapse. A number of specific results follow from our modeling and analysis. (1) Lateral membrane diffusion alone is insufficient as a mechanism for delivering AMPA receptors from the soma to distal dendrites. (2) A source of surface receptors at the soma tends to generate an exponential-like distribution of receptors along the dendrite, which has implications for synaptic democracy. (3) Diffusion mediates a heterosynaptic interaction between spines so that local changes in the constitutive recycling of AMPA receptors induce nonlocal changes in synaptic strength. On the other hand, structural changes in a spine following long term potentiation or depression have a purely local effect on synaptic strength. (4) A global change in the rates of AMPA receptor exo/endocytosis is unlikely to be the sole mechanism for homeostatic synaptic scaling. (5) The dynamics of AMPA receptor trafficking occurs on multiple timescales and varies according to spatial location along the dendrite. Understanding such dynamics is important when interpreting data from inactivation experiments that are used to infer the rate of relaxation to steady-state.

Diffusion-trapping model of receptor trafficking in dendrites.

We present a model for the diffusive trafficking of protein receptors along the surface of a neuron's dendrite. Distributed along the dendrite are spatially localized trapping regions that represent submicrometer mushroom-like protrusions known as dendritic spines. Within these trapping regions receptors can be internalized via endocytosis and either reinserted into the surface via exocytosis or degraded. We calculate the steady-state distribution of receptors along the dendrite assuming a constant flux of receptors inserted at one end, adjacent to the soma where receptors are synthesized, and use this to investigate the effectiveness of membrane diffusion as a transport mechanism. We also calculate the mean first passage time of a receptor to travel a certain distance along the cable and use this to derive an effective surface diffusivity.

Breathers in two-dimensional neural media.

In this Letter we show how nontrivial forms of spatially localized oscillations or breathers can occur in two-dimensional excitable neural media with short-range excitation and long-range inhibition. The basic dynamical mechanism involves a Hopf bifurcation of a stationary pulse solution in the presence of a spatially localized input. Such an input could arise from external stimuli or reflect changes in the excitability of local populations of neurons as a precursor for epileptiform activity. The resulting dynamical instability breaks the underlying radial symmetry of the stationary pulse, leading to the formation of a nonradially symmetric breather. The number of breathing lobes is consistent with the order of the dominant unstable Fourier mode associated with perturbations of the stationary pulse boundary.

Oscillatory waves in inhomogeneous neural media.

In this Letter we show that an inhomogeneous input can induce wave propagation failure in an excitatory neural network due to the pinning of a stationary front or pulse solution. A subsequent reduction in the strength of the input can lead to a Hopf instability of the stationary solution resulting in breatherlike oscillatory waves.

Saltatory waves in the spike-diffuse-spike model of active dendritic spines.

In this Letter we present the explicit construction of a saltatory traveling pulse of nonconstant profile in an idealized model of dendritic tissue. Excitable dendritic spine clusters, modeled with integrate-and-fire (IF) units, are connected to a passive dendritic cable at a discrete set of points. The saltatory nature of the wave is directly attributed to the breaking of translation symmetry in the cable. The conditions for propagation failure are presented as a function of cluster separation and IF threshold.

Effects of quasiactive membrane on multiply periodic traveling waves in integrate-and-fire systems.

We consider the dynamics of a one-dimensional continuum of synaptically interacting integrate-and-fire neurons with realistic forms of axodendritic interaction. The speed and stability of traveling waves are investigated as a function of discrete communication delays, distributed synaptic delays, and axodendritic delays arising from the spatially extended nature of the model neuron. In particular, dispersion curves for periodic traveling waves are constructed. Nonlinear ionic channels in the dendrite responsible for a so-called quasiactive bandpass response are shown to significantly influence the shape of dispersion curves. Moreover, a kinematic theory of spike train propagation suggests that period-doubling bifurcations of a singly periodic wave can occur in dendritic systems with a quasiactive membrane. The explicit construction of period-doubled solutions is used to confirm this prediction.

Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex.

This paper is concerned with a striking visual experience: that of seeing geometric visual hallucinations. Hallucinatory images were classified by Klüver into four groups called form constants comprising (i) gratings, lattices, fretworks, filigrees, honeycombs and chequer-boards, (ii) cobwebs, (iii) tunnels, funnels, alleys, cones and vessels, and (iv) spirals. This paper describes a mathematical investigation of their origin based on the assumption that the patterns of connection between retina and striate cortex (henceforth referred to as V1)-the retinocortical map-and of neuronal circuits in V1, both local and lateral, determine their geometry. In the first part of the paper we show that form constants, when viewed in V1 coordinates, essentially correspond to combinations of plane waves, the wavelengths of which are integral multiples of the width of a human Hubel-Wiesel hypercolumn, ca. 1.33-2 mm. We next introduce a mathematical description of the large-scale dynamics of V1 in terms of the continuum limit of a lattice of interconnected hypercolumns, each of which itself comprises a number of interconnected iso-orientation columns. We then show that the patterns of interconnection in V1 exhibit a very interesting symmetry, i.e. they are invariant under the action of the planar Euclidean group E(2)-the group of rigid motions in the plane-rotations, reflections and translations. What is novel is that the lateral connectivity of V1 is such that a new group action is needed to represent its properties: by virtue of its anisotropy it is invariant with respect to certain shifts and twists of the plane. It is this shift-twist invariance that generates new representations of E(2). Assuming that the strength of lateral connections is weak compared with that of local connections, we next calculate the eigenvalues and eigenfunctions of the cortical dynamics, using Rayleigh-Schrödinger perturbation theory. The result is that in the absence of lateral connections, the eigenfunctions are degenerate, comprising both even and odd combinations of sinusoids in straight phi, the cortical label for orientation preference, and plane waves in r, the cortical position coordinate. 'Switching-on' the lateral interactions breaks the degeneracy and either even or else odd eigenfunctions are selected. These results can be shown to follow directly from the Euclidean symmetry we have imposed. In the second part of the paper we study the nature of various even and odd combinations of eigenfunctions or planforms, the symmetries of which are such that they remain invariant under the particular action of E(2) we have imposed. These symmetries correspond to certain subgroups of E(2), the so-called axial subgroups. Axial subgroups are important in that the equivariant branching lemma indicates that when a symmetrical dynamical system becomes unstable, new solutions emerge which have symmetries corresponding to the axial subgroups of the underlying symmetry group. This is precisely the case studied in this paper. Thus we study the various planforms that emerge when our model V1 dynamics become unstable under the presumed action of hallucinogens or flickering lights. We show that the planforms correspond to the axial subgroups of E(2), under the shift-twist action. We then compute what such planforms would look like in the visual field, given an extension of the retinocortical map to include its action on local edges and contours. What is most interesting is that, given our interpretation of the correspondence between V1 planforms and perceived patterns, the set of planforms generates representatives of all the form constants. It is also noteworthy that the planforms derived from our continuum model naturally divide V1 into what are called linear regions, in which the pattern has a near constant orientation, reminiscent of the iso-orientation patches constructed via optical imaging. The boundaries of such regions form fractures whose points of intersection correspond to the well-known 'pinwheels'. To complete the study we then investigate the stability of the planforms, using methods of nonlinear stability analysis, including Liapunov-Schmidt reduction and Poincaré-Lindstedt perturbation theory. We find a close correspondence between stable planforms and form constants. The results are sensitive to the detailed specification of the lateral connectivity and suggest an interesting possibility, that the cortical mechanisms by which geometric visual hallucinations are generated, if sited mainly in V1, are closely related to those involved in the processing of edges and contours.

Dynamical mechanism for sharp orientation tuning in an integrate-and-fire model of a cortical hypercolumn.

Orientation tuning in a ring of pulse-coupled integrate-and-fire (IF) neurons is analyzed in terms of spontaneous pattern formation. It is shown how the ring bifurcates from a synchronous state to a non-phase-locked state whose spike trains are characterized by clustered but irregular fluctuations of the interspike intervals (ISIs). The separation of these clusters in phase space results in a localized peak of activity as measured by the time-averaged firing rate of the neurons. This generates a sharp orientation tuning curve that can lock to a slowly rotating, weakly tuned external stimulus. Under certain conditions, the peak can slowly rotate even to a fixed external stimulus. The ring also exhibits hysteresis due to the subcritical nature of the bifurcation to sharp orientation tuning. Such behavior is shown to be consistent with a corresponding analog version of the IF model in the limit of slow synaptic interactions. For fast synapses, the deterministic fluctuations of the ISIs associated with the tuning curve can support a coefficient of variation of order unity.

A phase model of temperature-dependent mammalian cold receptors.

We present a tractable stochastic phase model of the temperature sensitivity of a mammalian cold receptor. Using simple linear dependencies of the amplitude, frequency, and bias on temperature, the model reproduces the experimentally observed transitions between bursting, beating, and stochastically phase-locked firing patterns. We analyze the model in the deterministic limit and predict, using a Strutt map, the number of spikes per burst for a given temperature. The inclusion of noise produces a variable number of spikes per burst and also extends the dynamic range of the neuron, both of which are analyzed in terms of the Strutt map. Our analysis can be readily applied to other receptors that display various bursting patterns following temperature changes.

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.

Dynamics of strongly-coupled spiking neurons.

We present a dynamical theory of integrate-and-fire neurons with strong synaptic coupling. We show how phase-locked states that are stable in the weak coupling regime can destabilize as the coupling is increased, leading to states characterized by spatiotemporal variations in the interspike intervals (ISIs). The dynamics is compared with that of a corresponding network of analog neurons in which the outputs of the neurons are taken to be mean firing rates. A fundamental result is that for slow interactions, there is good agreement between the two models (on an appropriately defined timescale). Various examples of desynchronization in the strong coupling regime are presented. First, a globally coupled network of identical neurons with strong inhibitory coupling is shown to exhibit oscillator death in which some of the neurons suppress the activity of others. However, the stability of the synchronous state persists for very large networks and fast synapses. Second, an asymmetric network with a mixture of excitation and inhibition is shown to exhibit periodic bursting patterns. Finally, a one-dimensional network of neurons with long-range interactions is shown to desynchronize to a state with a spatially periodic pattern of mean firing rates across the network. This is modulated by deterministic fluctuations of the instantaneous firing rate whose size is an increasing function of the speed of synaptic response.

Resonantlike synchronization and bursting in a model of pulse-coupled neurons with active dendrites.

We analyze the dynamical effects of active, linearized dendritic membranes on the synchronization properties of neuronal interactions. We show that a pair of pulse-coupled integrate-and-fire neurons interacting via active dendritic cables can exhibit resonantlike synchronization when the frequency of the oscillators is approximately matched to the resonant frequency of the membrane impedance. For weak coupling the neurons are phase-locked with constant interspike intervals whereas for strong coupling periodic bursting patterns are observed. This bursting behavior is reflected by the occurrence of a Hopf bifurcation in the firing rates of a corresponding rate-coded model.

Mode locking and Arnold tongues in integrate-and-fire neural oscillators.

An analysis of mode-locked solutions that may arise in periodically forced integrate-and-fire (IF) neural oscillators is introduced based upon a firing map formulation of the dynamics. A q:p mode-locked solution is identified with a spike train in which p firing events occur in a period qDelta, where Delta is the forcing period. A linear stability analysis of the map of firing times around such solutions allows the determination of the Arnold tongue structure for regions in parameter space where stable solutions exist. The analysis is verified against direct numerical simulations for both a sinusoidally forced IF system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current. This approach is extended to the case of two synaptically coupled IF oscillators, showing that mode-locked states can exist for some self-consistently determined common period of repetitive firing. Numerical simulations show that such solutions have a bursting structure where regions of spiking activity are interspersed with quiescent periods before repeating. The influence of the synaptic current upon the Arnold tongue structure is explored in the regime of weak coupling.

Mean-field theory of globally coupled integrate-and-fire neural oscillators with dynamic synapses.

We analyze the effects of synaptic depression or facilitation on the existence and stability of the splay or asynchronous state in a population of all-to-all, pulse-coupled neural oscillators. We use mean-field techniques to derive conditions for the local stability of the splay state and determine how stability depends on the degree of synaptic depression or facilitation. We also consider the effects of noise. Extensions of the mean-field results to finite networks are developed in terms of the nonlinear firing time map.

Nonlinear shunting model of the pupil light reflex.

A leaky-integrator equation with shunting is introduced to represent processing at the retinal level as part of a system of delay-differential equations modelling the pupil light reflex. This system of equations exhibits the correct behaviour in response to monocular and binocular sinusoidal inputs. Understanding such behaviour can help to establish (non-invasively) the location and nature of nonlinearities in the reflex arc and how signals from the two eyes combine.

Integro-differential equations and the stability of neural networks with dendritic structure.

We analyse the effects of dendritic structure on the stability of a recurrent neural network in terms of a set of coupled, non-linear Volterra integro-differential equations. These, which describe the dynamics of the somatic membrane potentials, are obtained by eliminating the dendritic potentials from the underlying compartmental model or cable equations. We then derive conditions for Turing-like instability as a precursor for pattern formation in a spatially organized network. These conditions depend on the spatial distribution of axo-dendritic connections across the network.