NEURONAL DYNAMICS I:ACTIVATIONS AND SIGNALS NEURONS AS FUNCTIONS Neurons behave as functions. Neurons transduce an unbounded input activation x(t)at time t into a bounded output signal S(x(t)). 2
2 NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONS AS FUNCTIONS Neurons behave as functions. Neurons transduce an unbounded input activation x(t) at time t into a bounded output signal S(x(t))
NEURONAL DYNAMICS I:ACTIVATIONS AND SIGNALS NEURONS AS FUNCTIONS The transduction description:a sigmoidal or S-shaped curve the logistic signal function: S(x)= +e-a =eS(1-S)>0 (c>0) dx The logistic signal function is sigmoidal and strictly increases for positive scaling constant c>0. 3
3 NEURONS AS FUNCTIONS The transduction description: a sigmoidal or S-shaped curve the logistic signal function: cx e S x − + = 1 1 ( ) ' = = cS(1− S) 0 (c 0) dx dS S The logistic signal function is sigmoidal and strictly increases for positive scaling constant c >0. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS
NEURONAL DYNAMICS I:ACTIVATIONS AND SIGNALS NEURONS AS FUNCTIONS S(x) X -00 0 +00 Fig.1 s(x)is a bounded monotone-nondecreasing function of x If c+o,we get threshold signal function (dash line), Which is piecewise differentiable 4
4 NEURONS AS FUNCTIONS S(x) x -∞ - 0 + +∞ Fig.1 s(x) is a bounded monotone-nondecreasing function of x If c→+∞，we get threshold signal function (dash line), Which is piecewise differentiable NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS
NEURONAL DYNAMICS I:ACTIVATIONS AND SIGNALS SIGNAL MONOTONICITY In general,signal functions are monotone nondecreasing S'>=0.This means signal functions have an upper bound or saturation value. The staircase signal function is a piecewise-differentiable Monotone-nondecreasing signal function. 5
5 SIGNAL MONOTONICITY In general, signal functions are monotone nondecreasing S’>=0. This means signal functions have an upper bound or saturation value. The staircase signal function is a piecewise-differentiable Monotone-nondecreasing signal function. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS
NEURONAL DYNAMICS I:ACTIVATIONS AND SIGNALS SIGNAL MONOTONICITY An important exception:bell-shaped signal function or Gaussian signal functions S(x)=e-cx c>0 S=-2cxe S S"o∝-x The sign of the signal-activation derivation s'is opposite the sign of the activation x.We shall assume signal functions are monotone nondecreasing unless stated otherwise. 6
6 SIGNAL MONOTONICITY An important exception: bell-shaped signal function or Gaussian signal functions 0 2 = − S x e c cx ( ) S cxe S x cx = − − − ' 2 , ' 2 The sign of the signal-activation derivation s’ is opposite the sign of the activation x. We shall assume signal functions are monotone nondecreasing unless stated otherwise. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS