Stochastic neural field model of stimulus-dependent variability in cortical neurons

Dynamics of neural fields with exponential temporal kernel

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.

Response sub-additivity and variability quenching in visual cortex

Sub-additivity and variability are ubiquitous response motifs in the primary visual cortex (V1). Response sub-additivity enables the construction of useful interpretations of the visual environment, whereas response variability indicates the factors that limit the precision with which the brain can do this. There is increasing evidence that experimental manipulations that elicit response sub-additivity often also quench response variability. Here, we provide an overview of these phenomena and suggest that they may have common origins. We discuss empirical findings and recent model-based insights into the functional operations, computational objectives and circuit mechanisms underlying V1 activity. These different modelling approaches all predict that response sub-additivity and variability quenching often co-occur. The phenomenology of these two response motifs, as well as many of the insights obtained about them in V1, generalize to other cortical areas. Thus, the connection between response sub-additivity and variability quenching may be a canonical motif across the cortex.

Size tuning of neural response variability in laminar circuits of macaque primary visual cortex

A defining feature of the cortex is its laminar organization, which is likely critical for cortical information processing. For example, visual stimuli of different size evoke distinct patterns of laminar activity. Visual information processing is also influenced by the response variability of individual neurons and the degree to which this variability is correlated among neurons. To elucidate laminar processing, we studied how neural response variability across the layers of macaque primary visual cortex is modulated by visual stimulus size. Our laminar recordings revealed that single neuron response variability and the shared variability among neurons are tuned for stimulus size, and this size-tuning is layer-dependent. In all layers, stimulation of the receptive field (RF) reduced single neuron variability, and the shared variability among neurons, relative to their pre-stimulus values. As the stimulus was enlarged beyond the RF, both single neuron and shared variability increased in supragranular layers, but either did not change or decreased in other layers. Surprisingly, we also found that small visual stimuli could increase variability relative to baseline values. Our results suggest multiple circuits and mechanisms as the source of variability in different layers and call for the development of new models of neural response variability.

Self-organized criticality in a mesoscopic model of excitatory-inhibitory neuronal populations by short-term and long-term synaptic plasticity

Dynamics of an interconnected population of excitatory and inhibitory spiking neurons wandering around a Bogdanov-Takens (BT) bifurcation point can generate the observed scale-free avalanches at the population level and the highly variable spike patterns of individual neurons. These characteristics match experimental findings for spontaneous intrinsic activity in the brain. In this paper, we address the mechanisms causing the system to get and remain near this BT point. We propose an effective stochastic neural field model which captures the dynamics of the mean-field model. We show how the network tunes itself through local long-term synaptic plasticity by STDP and short-term synaptic depression to be close to this bifurcation point. The mesoscopic model that we derive matches the directed percolation model at the absorbing state phase transition.

Noise-driven bifurcations in a neural field system modelling networks of grid cells

The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by analysing the robustness of network activity patterns with respect to noise in an upscaled noisy grid cell model in the form of a system of partial differential equations. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We show that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.

Stochastic resonance and 'gamma band' synchronization in the human visual system

Cortical synchronization in the gamma-frequency range (above ~30.0 Hz) and the signal/noise interplay described by stochastic resonance models have been proposed as basic mechanisms in neuronal synchronization and sensory information processing, particularly in vision. Here we report an observation in humans of linear and inverted-U distributions of the electrophysiological (EEG) responses to visual contrast stimulation in the gamma band and in the low frequency components of the visual evoked responses (VER), respectively. The combination of linear and inverted-U distributions is described by a stochastic resonance model (SR). The observation needs replication in larger subjects' samples. It nevertheless adds to the available evidence of a role of gamma oscillatory signals and SR mechanisms in neuronal synchronization and visual processing. Some functional adaptation in human vision appears conceivable and further investigation is warranted.

Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses

Analytical forms for neuronal firing rates are important theoretical tools for the analysis of network states. Since the 1960s, the majority of approaches have treated neurons as being electrically compact and therefore isopotential. These approaches have yielded considerable insight into how single-cell properties affect network activity; however, many neuronal classes, such as cortical pyramidal cells, are electrically extended objects. Calculation of the complex flow of electrical activity driven by stochastic spatio-temporal synaptic input streams in these structures has presented a significant analytical challenge. Here we demonstrate that an extension of the level-crossing method of Rice, previously used for compact cells, provides a general framework for approximating the firing rate of neurons with spatial structure. Even for simple models, the analytical approximations derived demonstrate a surprising richness including: independence of the firing rate to the electrotonic length for certain models, but with a form distinct to the point-like leaky integrate-and-fire model; a non-monotonic dependence of the firing rate on the number of dendrites receiving synaptic drive; a significant effect of the axonal and somatic load on the firing rate; and the role that the trigger position on the axon for spike initiation has on firing properties. The approach necessitates only calculating the mean and variances of the non-thresholded voltage and its rate of change in neuronal structures subject to spatio-temporal synaptic fluctuations. The combination of simplicity and generality promises a framework that can be built upon to incorporate increasing levels of biophysical detail and extend beyond the low-rate firing limit treated in this paper.

Stochastic neural fields as gradient dynamical systems

Continuous attractor neural networks are used extensively to model a variety of experimentally observed coherent brain states, ranging from cortical waves of activity to stationary activity bumps. The latter are thought to play an important role in various forms of neural information processing, including population coding in primary visual cortex (V1) and working memory in prefrontal cortex. However, one limitation of continuous attractor networks is that the location of the peak of an activity bump (or wave) can diffuse due to intrinsic network noise. This reflects marginal stability of bump solutions with respect to the action of an underlying continuous symmetry group. Previous studies have used perturbation theory to derive an approximate stochastic differential equation for the location of the peak (phase) of the bump. Although this method captures the diffusive wandering of a bump solution, it ignores fluctuations in the amplitude of the bump. In this paper, we show how amplitude fluctuations can be analyzed by reducing the underlying stochastic neural field equation to a finite-dimensional stochastic gradient dynamical system that tracks the stochastic motion of both the amplitude and phase of bump solutions. This allows us to derive exact expressions for the steady-state probability density and its moments, which are then used to investigate two major issues: (i) the input-dependent suppression of neural variability and (ii) noise-induced transitions to bump extinction. We develop the theory by considering the particular example of a ring attractor network with SO(2) symmetry, which is the most common architecture used in attractor models of working memory and population tuning in V1. However, we also extend the analysis to a higher-dimensional spherical attractor network with SO(3) symmetry which has previously been proposed as a model of orientation and spatial frequency tuning in V1. We thus establish how a combination of stochastic analysis and group theoretic methods provides a powerful tool for investigating the effects of noise in continuous attractor networks.

Integration of cortical population signals for visual perception

Visual stimuli evoke heterogeneous responses across nearby neural populations. These signals must be locally integrated to contribute to perception, but the principles underlying this process are unknown. Here, we exploit the systematic organization of orientation preference in macaque primary visual cortex (V1) and perform causal manipulations to examine the limits of signal integration. Optogenetic stimulation and visual stimuli are used to simultaneously drive two neural populations with overlapping receptive fields. We report that optogenetic stimulation raises firing rates uniformly across conditions, but improves the detection of visual stimuli only when activating cells that are preferentially-tuned to the visual stimulus. Further, we show that changes in correlated variability are exclusively present when the optogenetically and visually-activated populations are functionally-proximal, suggesting that correlation changes represent a hallmark of signal integration. Our results demonstrate that information from functionally-proximal neurons is pooled for perception, but functionally-distal signals remain independent.