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Exploiting the Hamiltonian structure of a neural field model. (English) Zbl 1203.37034

This work is concerned with the following model \[ \frac{\partial u(x,t)}{\partial t} \, = \, - u(x,t) + \int_{-\infty}^{+\infty} w(x-y) f[u(y,t)] dy, \] where \(u(x,t)\) is the average voltage, or activity level, of a neuronal population of spatial position \(x\) and time \(t\). The coupling function \(w(x)\) is the distance-dependent strength of connectivity between neuronal elements and is given by \[ w(x) \, = \, e^{-b|x|} \left( b \sin |x| + \cos x \right), \] with \(b\) a parameter which governs the rate at which oscillations in \(w\) decay with distance. The firing rate function \(f(u)\) models neurons firing once threshold is reached and tends to a maximal limit as the stimulus is increased. It is given by: \[ f(u) \, = \, 2 \, \exp \left[-r/(u-\theta)^2 \right] \Theta(u-\theta), \] where the parameter \(\theta\) is the firing threshold, \(r\) is the steepness parameter and \(\Theta\) is the Heaviside step function. We recall that the Heaviside step function is defined by \[ \Theta(x) \, = \, \int_{-\infty}^{x} \delta(t) \, dt, \] where \(\delta(t)\) is the Dirac delta function. \(\Theta(x)\) is zero for \(x<0\) and it is \(1\) for \(x>0\). This model has been studied in [C. R. Laing et al., SIAM J. Appl. Math., 63, No.1, 62–97 (2002; Zbl 1017.45006)] and [C. R. Laing, W.C. Troy, SIAM J. Appl. Dyn. Syst., 2, No. 3, 487–516 (2003; Zbl 1088.34011)].
The authors consider time-independent solutions of the equation which are stationary and spatially-localised, that is, \[ \lim_{|x| \to \infty} (u,u',u'',u''') \, = \, (0,0,0,0). \] These solutions satisfy the integral equation \[ u(x) \, = \, \int_{-\infty}^{\infty} w(x-y) f[u(y)] dy, \] which can be transformed to the following differential equation by use of Fourier transforms: \[ u'''' - 2(b^2-1) u'' + (b^2+1)^2 u \, = \, 4b(b^2+1) f(u). \tag{eq9} \] This equation can be written as a system of four first-order ODEs with the following properties:
1) the linearisation of the system at the origin has the four eigenvalues \(\pm b\pm i\), i.e., the origin is a bifocus with a two-dimensional unstable manifold and a two-dimensional stable manifold.
2) the system can be written as a Hamiltonian system.
3) this Hamiltonian system is invariant under a space-reversing symmetry.
Using all these properties the authors derive a real scalar function whose zeros give rise to the single-bump homoclinic orbits to the fixed point at the origin of (eq9).
The authors numerically integrate the equations so as to examine the unexpected disappearance of stable homoclinic orbits in certain regions of the parameter space. The solution curve of stable homoclinic orbits breaks when the firing rate function is sufficiently steep.

MSC:

37C29 Homoclinic and heteroclinic orbits for dynamical systems
92C20 Neural biology

Software:

HomCont; AUTO
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References:

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