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In volume transmission (or neuromodulation) neurons do not make one-to-one connections to other neurons, but instead simply release neurotransmitter into the extracellular space from numerous varicosities. Many well-known neurotransmitters including serotonin (5HT), dopamine (DA), histamine (HA), Gamma-Aminobutyric Acid (GABA) and acetylcholine (ACh) participate in volume transmission. Typically, the cell bodies are in one volume and the axons project to a distant volume in the brain releasing the neurotransmitter there. We introduce volume transmission and describe mathematically two natural homeostatic mechanisms. In some brain regions several neurotransmitters in the extracellular space affect each other's release. We investigate the dynamics created by this comodulation in two different cases: serotonin and histamine; and the comodulation of 4 neurotransmitters in the striatum and we compare to experimental data. This kind of comodulation poses new dynamical questions as well as the question of how these biochemical networks influence the electrophysiological networks in the brain. [ABSTRACT FROM AUTHOR]

The qualitative behavior of the differential equation x ′ = f (t , x) , where x is real and f is periodic in t, is determined by the attracting periodic solutions. This article gives an exposition of a method of Ilyashenko that uses complex analysis to determine an upper bound on the number of periodic solutions. The method is then applied to a well-known model of a single neuron or pool of neurons. [ABSTRACT FROM AUTHOR]

In this paper, a recurrent neural network with mixed delays which plays an important role is considered. We are concerned with the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions. First, we establish results that are interesting on the functional space of such functions like composition theorem. Second, by employing the fixed-point theorem and some properties of the measure pseudo almost automorphic functions, some sufficient conditions for the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions have been established. Our results obtained in this paper are new. Finally, two commonly used numerical examples are given to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Subthreshold oscillations in neurons are those oscillations which do not attain the critical value of the membrane's voltage needed for triggering an action potential (a spike). Their contribution to the forming of action potentials in neurons is a current field of research in biology. The present work approaches this subject using tools from mathematical modelling, more exactly, a neuronal non-smooth map-based model is proposed and studied. The behaviour of the model in a noisy medium is also studied. [ABSTRACT FROM AUTHOR]

Exact, steady-state, single-front solutions are constructed for a spatially discrete bistable equation with a piecewise linear reaction term, known as a sawtooth nonlinearity. These solutions are obtained by solving second-order difference equations with variable coefficients, which are linear under certain assumptions on the expected solutions. An algorithmic procedure for constructing solutions in general, for both homogeneous and inhomogeneous diffusion, is obtained using a combination of Jacobi-Operator theory and the Sherman–Morrison formula. The existence of solutions for the difference equation, implies propagation failure of fronts for the corresponding differential-difference equation. The interval of propagation failure, which is the range of values of the detuning parameter that render stationary fronts, is studied in detail for the case of a single defect in the medium of propagation. Explicit formulae reveal precise relationships between parameter values that cause traveling fronts to fail to propagate when the interface reaches the inhomogeneities in the medium. These explicit formulae are also compared to numerical computations using the proposed algorithmic approach, which provides a check of its computational usefulness and illustrates its capabilities for problems with more complicated choices of parameter values. [ABSTRACT FROM PUBLISHER]