Journal of Mathematical Neuroscience

Coding for visual stimuli in the ventral stream is known to be invariant to object identity preserving nuisance transformations. Indeed, much recent theoretical and experimental work suggests that the main challenge for the visual cortex is to build up such nuisance invariant representations. Recently, artificial convolutional networks have succeeded in both learning such invariant properties and, surprisingly, predicting cortical responses in macaque and mouse visual cortex with unprecedented accuracy. However, some of the key ingredients that enable such success-supervised learning and the backpropagation algorithm-are neurally implausible...

In this paper, we present a novel model of the primary visual cortex (V1) based on orientation, frequency, and phase selective behavior of V1 simple cells. We start from the first-level mechanisms of visual perception, receptive profiles. The model interprets V1 as a fiber bundle over the two-dimensional retinal plane by introducing orientation, frequency, and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as rotated, frequency modulated, and phase shifted Gabor function...

Binocular rivalry occurs when the two eyes are presented with incompatible stimuli and perception alternates between these two stimuli. This phenomenon has been investigated in two types of experiments: (1) Traditional experiments where the stimulus is fixed, (2) eye-swap experiments in which the stimulus periodically swaps between eyes many times per second (Logothetis et al. in Nature 380(6575):621-624, 1996). In spite of the rapid swapping between eyes, perception can be stable for many seconds with specific stimulus parameter configurations...

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction...

Networks of neurons in the cerebral cortex exhibit a balance between excitation (positive input current) and inhibition (negative input current). Balanced network theory provides a parsimonious mathematical model of this excitatory-inhibitory balance using randomly connected networks of model neurons in which balance is realized as a stable fixed point of network dynamics in the limit of large network size. Balanced network theory reproduces many salient features of cortical network dynamics such as asynchronous-irregular spiking activity...

This is the first half of a two-part paper dealing with the geometry of color perception. Here we analyze in detail the seminal 1974 work by H.L. Resnikoff, who showed that there are only two possible geometric structures and Riemannian metrics on the perceived color space [Formula: see text] compatible with the set of Schrödinger's axioms completed with the hypothesis of homogeneity. We recast Resnikoff's model into a more modern colorimetric setting, provide a much simpler proof of the main result of the original paper, and motivate the need of psychophysical experiments to confute or confirm the linearity of background transformations, which act transitively on [Formula: see text]...

Following publication of the original article (Naud and Longtin in J Math Neurosci 9:3, 2019), the authors noticed a mistake in the first paragraph within "Altered propagation".

Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory for mesoscopic population dynamics with static synapses to the case of dynamic synapses exhibiting short-term plasticity (STP). The extended theory offers an approximate mean-field dynamics for the synaptic input currents arising from populations of spiking neurons and synapses undergoing Tsodyks-Markram STP...

Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Constant stimulation can lead to side effects, it is therefore desirable to find ways to stimulate less while maintaining clinical efficacy. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor. To advance methods to optimise deep brain stimulation while providing insights into tremor circuits, we ask the question: can the effects of phase-locked stimulation be accounted for by a canonical Wilson-Cowan model? We first analyse patient data, and identify in half of the datasets significant dependence of the effects of stimulation on the phase at which stimulation is provided...

The fruit fly's natural visual environment is often characterized by light intensities ranging across several orders of magnitude and by rapidly varying contrast across space and time. Fruit fly photoreceptors robustly transduce and, in conjunction with amacrine cells, process visual scenes and provide the resulting signal to downstream targets. Here, we model the first step of visual processing in the photoreceptor-amacrine cell layer. We propose a novel divisive normalization processor (DNP) for modeling the computation taking place in the photoreceptor-amacrine cell layer...

The modeling of neural fields in the visual cortex involves geometrical structures which describe in mathematical formalism the functional architecture of this cortical area. The case of contour detection and orientation tuning has been extensively studied and has become a paradigm for the mathematical analysis of image processing by the brain. Ten years ago an attempt was made to extend these models by replacing orientation (an angle) with a second-order tensor built from the gradient of the image intensity, and it was named the structure tensor...

Neurons are biological cells with uniquely complex dendritic morphologies that are not present in other cell types. Electrical signals in a neuron with branching dendrites can be studied by cable theory which provides a general mathematical modelling framework of spatio-temporal voltage dynamics. Typically such models need to be solved numerically unless the cell membrane is modelled either by passive or quasi-active dynamics, in which cases analytical solutions can be reduced to calculation of the Green's function describing the fundamental input-output relationship in a given morphology...

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regression problem (Harris et al. in Neural Information Processing Systems, 2016), but existing techniques do not scale to the whole-brain setting. The corresponding matrix equation is challenging to solve due to large scale, ill-conditioning, and a general form that lacks a convergent splitting. We propose a greedy low-rank algorithm for the connectome reconstruction problem in very high dimensions...

Following publication of the original article [1], the authors noticed a mistake in the first paragraph within "Altered propagation".

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter ϵ uncouples the system at [Formula: see text]. Using a normal form for [Formula: see text] identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson-Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in the existing models)...

The emergent activity of biological systems can often be represented as low-dimensional, Langevin-type stochastic differential equations. In certain systems, however, large and abrupt events occur and violate the assumptions of this approach. We address this situation here by providing a novel method that reconstructs a jump-diffusion stochastic process based solely on the statistics of the original data. Our method assumes that these data are stationary, that diffusive noise is additive, and that jumps are Poisson...

The canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift-diffusion model (DDM). In this model, a decision variable keeps track of the integrated difference in sensory evidence for two competing alternatives. Here I extend the notion of a drift-diffusion process to multiple alternatives. The competition between n alternatives takes place in a linear subspace of [Formula: see text] dimensions; that is, there are [Formula: see text] decision variables, which are coupled through correlated noise sources...

BACKGROUND: Trigeminal neuralgia (TN) is a severe neuropathic pain, which has an electric shock-like characteristic. There are some common treatments for this pain such as medicine, microvascular decompression or radio frequency. In this regard, transcranial direct current stimulation (tDCS) is another therapeutic method to reduce pain, which has been recently attracting the therapists' attention. The positive effect of tDCS on TN was shown in many previous studies. However, the mechanism of the tDCS effect has remained unclear...

To establish and exploit novel biomarkers of demyelinating diseases requires a mechanistic understanding of axonal propagation. Here, we present a novel computational framework called the stochastic spike-diffuse-spike (SSDS) model for assessing the effects of demyelination on axonal transmission. It models transmission through nodal and internodal compartments with two types of operations: a stochastic integrate-and-fire operation captures nodal excitability and a linear filtering operation describes internodal propagation...

Understanding nervous system function requires careful study of transient (non-equilibrium) neural response to rapidly changing, noisy input from the outside world. Such neural response results from dynamic interactions among multiple, heterogeneous brain regions. Realistic modeling of these large networks requires enormous computational resources, especially when high-dimensional parameter spaces are considered. By assuming quasi-steady-state activity, one can neglect the complex temporal dynamics; however, in many cases the quasi-steady-state assumption fails...