Threshold Curve for the Excitability of Bidimensional Spiking Neurons

International audienceWe shed light on the threshold for spike initiation in two-dimensional neuron models. A threshold criterion that depends on both the membrane voltage and the recovery variable is proposed. This approach provides a simple and unified framework that accounts for numerous voltage threshold properties including adaptation, variability and time-dependent dynamics. In addition, neural features such as accommodation, inhibition-induced spike, and post-inhibitory (-excitatory) facilitation are the direct consequences of the existence of a threshold curve. Implications for neural modeling are also discussed

Bifurcation properties of the average activity of interconnected neural populations

Abstract.: The relevant scale for the study of the electrical activity of neural networks is a problem of mathematical and biological interest. From a continuous model of the cortex activity we derive a simple model of an interconnected pair of excitatory and inhibitory neural populations that describes the activity of a homogeneous network. Our model depends on three parameters that stand for the scale variability of the network. A bifurcation analysis reveals a great variety of patterns that arise from the interplay of excitatory and inhibitory populations provided by synaptic interactions. We emphasize the differences between the dynamical regimes when considering a moderate and a high inhibitory scale. We discuss the consequences on a propagating activit

Stabilization of pulse waves through inhibition in a feedforward neural network

International audienceWe consider a firing rate model of a neural network of excitatory and inhibitory populations with an excitatory feedforward connectivity. We analyze traveling wave solutions and determine the conditions for their existence and stability. Our study demonstrates the role of inhibition in stable pulse propagation. In a purely excitatory network, pulse waves are unstable because of the existence of stable front wave and back wave with different velocities. Pulse waves can propagate stably in the network where excitation is appropriately balanced by inhibition. Analytical results on the wave speeds and the shape of waves are obtained

On the number of limit cycles in piecewise linear Liénard systems

International audienceIn a previous paper [Tonnelier, 2002] we conjectured that a Liénard system of the form x' = p(x) − y, y' = x where p is piecewise linear on n + 1 intervals has up to 2n limit cycles. We construct here a general class of functions p satisfying this conjecture. Limit cycles are obtained from the bifurcation of the linear center

Propagation of spike sequences in neural networks

International audiencePrecise spatiotemporal sequences of action potentials are observed in many brain areas and are thought to be involved in the neural processing of sensory stimuli. Here, we examine the ability of spiking neural networks to propagate stably a spatiotemporal sequence of spikes in the limit where each neuron fires only one spike. In contrast to previous studies on propagation in neural networks, we assume only homogeneous connectivity and do not use the continuum approximation. When the propagation is associated with a simple traveling wave, or a one-spike sequence, we derive some analytical results for the wave speed and show that its stability is determined by the Schur criterion. The propagation of a sequence of several spikes corresponds to the existence of stable composite waves, i.e., stable spatiotemporal periodic traveling waves. The stability of composite waves is related to the roots of a system of multivariate polynomials. Using the simplest synaptic architecture that supports composite waves, a three nearest-neighbor coupling feedforward network, we analytically and numerically investigate the propagation of 2-composite waves, i.e., two-spike sequence propagation. The influence of the synaptic coupling, stochastic perturbations, and neuron parameters on the propagation of larger sequences is also investigated

Signal propagations along excitable chains

International audienceA simplified continuous-time threshold model for wave propagation in excitable media is proposed. The ability of the resulting transmission line to convey simple signals is investigated. Existence and multistability of traveling waves where two successive units share the same waveform is established. We show that, depending on the connectivity of the transmission line, an arbitrary number of distinct signals can be transmitted. The influence of model parameters (time constants, coupling strength, and connectivity) on the traveling signal properties is analyzed

Categorization of neural excitability using threshold models

International audienceA classification of spiking neurons according to the transition from quiescence to periodic firing of action potentials is commonly used. Nonbursting neurons are classified into two types, type I and type II excitability. We use simple phenomenological spiking neuron models to derive a criterion for the determination of the neural excitability based on the afterpotential following a spike. The crucial characteristic is the existence for type II model of a positive overshoot, i.e. a delayed afterdepolarization, during the recovery process of the membrane potential. Our prediction is numerically tested using well known type I and type II models including the Connor et al. model and the Hodgkin-Huxley model

Cyclic negative feedback systems: what is the chance of oscillation?

International audienceMany biological oscillators have a cyclic structure consisting of negative feedback loops. In this paper, we analyze the impact that the addition of a positive or a negative self-feedback loop has on the oscillatory behaviour of the three negative feedback oscillators proposed by Tsai et al (Science 231:126-129, 2008) where, in contast with numerous oscillator models, the interactions between elements occur via the modulation of the degradation rates. Through analytical and computational studies we show that an additional self-feedback affects the dynamical behaviour. In the high cooperativity limit, i.e. for large Hill coefficients, we derive exact analytical conditions for oscillations and show that the relative location between the dissociation constants of the Hill functions and the ratio of kinetic parameters determines the possibility of oscillatory activities. We compute analytically the probability of oscillations for the three models and show that the smallest domain of periodic behaviour is obtained for the negative-plus-negative feedback system whereas the additional positive self-feedback loop does not modify significantly the chance to oscillate. We numerically investigate to what extent the properties obtained in the sharp situation applied in the smooth case. Results suggest that a switch-like coupling behaviour, a time-scale separation and a repressilator-type architecture with an even number of elements facilitate the emergence of sustained oscillations in biological systems. An additional positive self-feedback loop produces robustness and adaptability whereas an additional negative self-feedback loop reduces the chance to oscillate

Screening of chemicals for human bioaccumulative potential with a physiologically based toxicokinetic model

Human bioaccumulative potential is an important element in the risk assessment of chemicals. Due to the high number of synthetic chemicals, there exists the need to develop prioritisation strategies. The purpose of this study was to develop a predictive tool for human bioaccumulation risk assessment that incorporates not only the chemical properties of the compounds, but also the processes that tend to decrease the concentration of the compound such as metabolisation. We used a generic physiologically based toxicokinetic model that based on in vitro human liver metabolism data, minimal renal excretion and a constant exposure was able to assess the bioaccumulative potential of a chemical. The approach has been analysed using literature data on well-known bioaccumulative compounds and liver metabolism data from the ECVAM database and a subset of the ToxCast phase I chemical library—in total 94 compounds covering pharmaceuticals, plant protection products and industrial chemicals. Our results provide further evidence that partitioning properties do not allow for a reliable screening criteria for human chemical hazard. Our model, based on a 100% intestinal absorption assumption, suggests that metabolic clearance, plasma protein-binding properties and renal excretion are the main factors in determining whether bioaccumulation will occur and its amount. It is essential that in vitro metabolic clearance tests with metabolic competent cell lines as well as plasma protein-binding assays be performed for suspected bioaccumulative compounds

Solitary waves in the excitable Burridge-Knopoff Model

International audienceThe Burridge-Knopoff model describes the dynamics of an elastic chain of blocks pulled over a surface. This model accounts for nonlinear friction phenomena and displays excitability when the velocity-dependent friction force is non-monotone. We introduce a simplified piecewise linear friction law (reminiscent of the McKean nonlinearity in spiking neuron models) which allows us to analyze the existence of large amplitude solitary waves. Propagation failure is shown to occur for weakly coupled oscillators