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2001/02 Lecture 6. Examples of mathematical and computational models: Predator-prey models; Spike generation models; Associative memory models (discrete time); Neural oscillator models; Conservative Systems Newton’s law mx F (x ) . Here we have no damping or friction, and we can say that the energy is conserved (conservative system). dV Let V(x) denote the potential energy, F ( x ) . Then dx dV mx 0. dx dV d 1 1 x ( mx ) 0 mx 2 V ( x ) 0 E mx 2 V ( x ) const dx dt 2 2 Energy E is constant. Pendulum Lx g sin x 1 L x g m t where x is the angle from downward vertical, g is the acceleration due to gravity, L is the length of the pendulum. Dimensionless form: x sin x or x y y sin x Equilibrium points: x 0, y 0, 1, 2 i - Centre. x , y 0, 1 1, u1 (1,1); 2 1, u2 (1,1) - Saddle. Damping to the pendulum: x y y b * y sin x Hopfield Neural Network, Attractor Neural Network Let x(t) is binary vector (01 or +1); 1, if wij x j (t ) 0 x i (t 1) 1, if wij x j (t ) 0 x (t ), wij x j (t ) 0 wij w ji , wii 0 “Energy” E(t ) 1 / 2 wij xi x j Assynchronose dynamics: update only one unit per time. E(t 1) E(t ) 1 / 2( x1 (t 1) x1 (t )) wij x j 0 Associative memory: memorise vectors α, β, ... wij i j i j ... Continous time: xi xi wij g ( x j ) Ii , g () is the sigmoid function E 1 / 2 wij gi g j Ii gi Predator-Prey Models Bazykin’s model of a predator-prey ecosystem: xy 2 x x 1 x x y y xy y 1 x 2 x and y are (scaled) numbers of a prey and predator, respectively. , , , are nonnegative parameters describing the behaviour of isolated populations and their interaction. Three population model: E E EC EW C C EC W W EW where E(t) is the elk population, C(t) is the coyote population, W(t) is the wolf population. All populations are measured in thousands of animals. t is measured in years (from 1995). Oscillating Chemical Reactions 4 xy x a x 1 x 2 y bx1 y 2 1 x where x and y are the dimensioless concentrations, a and b are positive parameters. (See Appendix 1 bellow). Spike generation models Slow-fast equations: x y F ( x) y a kx Van der Pol equation x ax( x 2 1) x 0 also is an example of slow-fast equations. Notice that d x ax( x 2 1) [ x a( 13 x 3 x)] . dt So, if we let F ( x) 13 x 3 x, z x aF ( x) , the Van der Pol equation implies that z x ax( x 2 1) x Hence the Van der Pol equation may be rewritten as x z aF ( x) z x d d , 1 / a 2 then Let y z / a, dt d a 3 x a[ y F ( x)] 1 y a x x y F ( x) . y x Nullclines: y F ( x), x 0 . Equilibrium point is a point of the nullclines intersection. Vectors, vector field. Fast and slow time scales (movements). Relaxation oscillations. Spike generation effect: x x * (1 x )(1 x ) y y x a x(0) I , y (0) y0 where x is potential, I is external input, a is constant. Threshold. Hodgkin-Huxley model (See appendix 2 bellow). Neural oscillator model 1 E E f ( wee E wei I P) 2 I I f ( wie E wii I Q) where E(t) is the average activity of excitatory neural population, I(t) is the average activity of inhibitory neural population, w are the connection strengths, P and Q are the external inputs, f ( . ) is the sigmoid function. Lorenz equation (chaotic behaviour) x ( y x ) y rx y xz z xy bz where x, y, z are variable to describe convection in the fluid layer, is the Prandtl number, r is the Rayleigh number, b is related to the height of the fluid layer. (See Appendix 3 bellow). 4