Chapter 2.NeuralDynamics:Activationand Signalsx2.2 Signal MonotonicityAn important exception: Gaussian signal functionsS(x) = e-ar2c>0(3)S'= -2cxe-ar?(4)We shall assume signal functions are monotone nondecreasingunless statedotherwise
Chapter 2. Neural Dynamics:Activation and Signals ※2.2 Signal Monotonicity An important exception: Gaussian signal functions 0 2 = − S x e c cx ( ) (3) 2 2 cx S cxe − ' = − (4) We shall assume signal functions are monotone nondecreasing unless stated otherwise
Chapter 2.Neural Dynamics:Activationand SignalsX2.2 Signal MonotonicityPotential or radial basis function for generalized Gaussiansignal function:n1Z(xj2S,(x) = exp[(5)28?In whichX=(xi,,xn)eRn(6)u, =(ui,..,uh)(7)
Chapter 2. Neural Dynamics:Activation and Signals ※2.2 Signal Monotonicity Potential or radial basis function for generalized Gaussian signal function: ( ) = exp[− ( − ) ] (5) n j i j j i i S x x 2 2 2 1 In which n x = (x1 , , xn )R (6) ( , , ) (7) i n i i = 1
Chapter 2.NeuralDynamics:Activationand Signalsx2.2 Signal MonotonicityA property of signal monotonicity: semi-linearityComparation:a. Linear signal functions:computation and analysis is comparatively easy;do not suppress noise.b. Nonlinear signal functions:increases a network's computational richness andfacilitates noise suppression;risks computational and analytical intractability:favors dynamical instability
Chapter 2. Neural Dynamics:Activation and Signals ※2.2 Signal Monotonicity A property of signal monotonicity: semi-linearity Comparation: a. Linear signal functions: computation and analysis is comparatively easy; do not suppress noise. b. Nonlinear signal functions: increases a network’s computational richness and facilitates noise suppression; risks computational and analytical intractability; favors dynamical instability
Chapter 2.NeuralDynamics:Activationand Signalsx2.2 Signal MonotonicitySignal and activation velocities dS/dt ,denoted SdS dxsSx(8)dx dtSignal velocities depend explicitly on action velocities
Chapter 2. Neural Dynamics:Activation and Signals ※2.2 Signal Monotonicity Signal and activation velocities dS/dt ,denoted Signal velocities depend explicitly on action velocities S S x dt dx dx dS S = = ' (8)