We present a mathematical model of a neural mass developed by a number of people, including Lopes da Silva and Jansen. This model features three interacting populations of cortical neurons and is described by a six-dimensional nonlinear dynamical system. We address some aspects of its behavior through a bifurcation analysis with respect to the input parameter of the system. This leads to a compact description of the oscillatory behaviors observed in Jansen and Rit (1995) (alpha activity) and Wendling, Bellanger, Bartolomei, and Chauvel (2000) (spike-like epileptic activity). In the case of small or slow variation of the input, the model can even be described as a binary unit. Again using the bifurcation framework, we discuss the influence of other parameters of the system on the behavior of the neural mass model.

Neural fields are an interesting option for modelling macroscopic parts of the cortex involving several populations of neurons, like cortical areas. Two classes of neural field equations are considered: voltage and activity based. The spatio-temporal behaviour of these fields is described by nonlinear integro-differential equations. The integral term, computed over a compact subset of R q , q = 1, 2, 3, involves space and time varying, possibly non-symmetric, intra-cortical connectivity kernels. Contributions from white matter afferents are represented as external input. Sigmoidal nonlinearities arise from the relation between average membrane potentials and instantaneous firing rates. Using methods of functional analysis, we characterize the existence and uniqueness of a solution of these equations for general, homogeneous (i.e. independent of the spatial variable), and spatially locally homogeneous inputs. In all cases we give sufficient conditions on the connectivity functions for the solutions to be absolutely stable, that is to say asymptotically independent of the initial state of the field. These conditions bear on some compact operators defined from the connectivity kernels, the maximal slope of the sigmoids, and the time constants used in describing the temporal shape of the post-synaptic potentials. Numerical experiments are presented to illustrate the theory. To our knowledge this is the first time that such a complete analysis of the problem of the existence and uniqueness of a solution of these equations has been obtained. Another important contribution is the analysis of the absolute stability of these solutions, more difficult but more general than the linear stability analysis which it implies. The reason why we have been able to complete this work programme is our use of the functional analysis framework and the theory of compact operators in a Hilbert space which has allowed us to provide simple mathematical answers to some of the questions raised by modellers in neuroscience.

Neural continuum networks are an important aspect of the modeling of macroscopic parts of the cortex. Two classes of such networks are considered: voltage and activity based. In both cases, our networks contain an arbitrary number, n, of interacting neuron populations. Spatial nonsymmetric connectivity functions represent cortico-cortical, local connections, and external inputs represent nonlocal connections. Sigmoidal nonlinearities model the relationship between (average) membrane potential and activity. Departing from most of the previous work in this area, we do not assume the nonlinearity to be singular, that is, represented by the discontinuous Heaviside function. Another important difference from previous work is that we relax the assumption that the domain of definition where we study these networks is infinite, that is, equal to [Formula: see text] or [Formula: see text]. We explicitly consider the biologically more relevant case of a bounded subset Ω of [Formula: see text], a better model of a piece of cortex. The time behavior of these networks is described by systems of integro-differential equations. Using methods of functional analysis, we study the existence and uniqueness of a stationary (i.e., time-independent) solution of these equations in the case of a stationary input. These solutions can be seen as ‘persistent’; they are also sometimes called bumps. We show that under very mild assumptions on the connectivity functions and because we do not use the Heaviside function for the nonlinearities, such solutions always exist. We also give sufficient conditions on the connectivity functions for the solution to be absolutely stable, that is, independent of the initial state of the network. We then study the sensitivity of the solutions to variations of such parameters as the connectivity functions, the sigmoids, the external inputs, and, last but not least, the shape of the domain of existence Ω of the neural continuum networks. These theoretical results are illustrated and corroborated by a large number of numerical experiments in most of the cases 2 ⩽ n ⩽ 3, 2 ⩽ q ⩽ 3.

Precise connections in the brain result from elaborate processes during development. In the visual system, axonal projections from retinal ganglion cells (RGCs) onto the superior colliculus (SC) form a precise retinotopic map. Studies have revealed that the development of retinocollicular maps involves three main factors: graded expression of molecular guidance cues such as EphAs and ephrin-As, activity-dependent processes driven by spontaneous activity in RGCs, and different forms of axonal competition. In this study, we develop a new, versatile model including these factors. We first model the selective arborization of RGC axons, mediated by EphA/ephrin-A signaling, without assuming that this initial process instructs the map's final topology. We also derive an integro-differential equation modeling a second, dynamic phase where activity-dependent plasticity of axonal arbors combined with their competition for collicular resources can deeply remodel the topology of immature maps. Our model hence challenges the view that retinotopic maps are instructed by matching molecular gradients and then merely refined by activity-dependent processes. We reproduce fine features of retinototopic map development in wild type and various transgenic mice, allowing a new understanding of the underlying mechanisms. Our model predicts that competition is not based on comparisons of axonal EphA receptor levels, but rather relies on the optimization of collicular resources mediated by neurotrophic receptors such as p75NTR. Our model finally clarifies the elusive role of reverse signaling between retinal ephrin-As and collicular EphAs by reproducing for the first time the phenotypes of two mouse genotypes where this function is altered.

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