1 / 32

NEURAL NETWORK THEORY NEURONAL DYNAMICS Ⅰ : ACTIVATIONS AND SIGNALS

NEURAL NETWORK THEORY NEURONAL DYNAMICS Ⅰ : ACTIVATIONS AND SIGNALS. 欢迎大家提出意见建议! 2003.10.15. NEURONAL DYNAMICS Ⅰ : ACTIVATIONS AND SIGNALS. NEURONS AS FUNCTIONS. Neurons behave as functions.

india
Download Presentation

NEURAL NETWORK THEORY NEURONAL DYNAMICS Ⅰ : ACTIVATIONS AND SIGNALS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. NEURAL NETWORK THEORY NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS • 欢迎大家提出意见建议! • 2003.10.15

  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)).

  3. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONS AS FUNCTIONS The transduction description: a sigmoidal or S-shaped curve the logistic signal function: The logistic signal function is sigmoidal and strictly increases for positive scaling constant c >0.

  4. -∞ - + +∞ 0 NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONS AS FUNCTIONS S(x) x 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

  5. 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.

  6. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL MONOTONICITY An important exception: bell-shaped signal function or Gaussian signal functions 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.

  7. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL MONOTONICITY Generalized Gaussian signal function define potential or radial basis function : input activation vector: variance: mean vector: we shall consider only scalar-input signal functions:

  8. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL MONOTONICITY neurons are nonlinear but not too much so ---- a property as semilinearity Linear signal functions - make computation and analysis comparatively easy - do not suppress noise - linear network are not robust Nonlinear signal functions - increases a network’s computational richness - increases a network’s facilitates noise suppression - risks computational and analytical intractability - favors dynamical instability

  9. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL MONOTONICITY Signal and activation velocities the signal velocity: =dS/dt Signal velocities depend explicitly on action velocities

  10. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS BIOLOGICAL ACTIVATIONS AND SIGNALS Fig.2 Neuron anatomy 神经元(Neuron)是由细胞核(cell nucleus),细胞体(soma),轴突(axon),树突(dendrites)和突触(synapse)所构成的

  11. x1 w1 x2 w2 net=XW o=f(net) ∑ f … xn wn NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS BIOLOGICAL ACTIVATIONS AND SIGNALS X=(x1,x2,…,xn) W=(w1,w2,…,wn) net=∑xiwi net=XW

  12. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS BIOLOGICAL ACTIVATIONS AND SIGNALS Competitive Neuronal Signal The neuron “wins” at time t if , “loses” if and otherwise possesses a fuzzy win-loss status between 0 an 1. a. Binary signal functions : [0,1] b. Bipolar signal functions : [-1,1] McCulloch—Pitts (M—P) neurons logical signal function ( Binary  Bipolar )

  13. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURON FIELDS Neurons within a field are topologically ordered, often by proximity. zeroth-order topology : lack of topological structure Denotation: , , neural system samples the function m times to generate the associated pairs , ... , The overall neural network behaves as an adaptive filter and sample data changed network parameters.

  14. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONAL DYNAMICAL SYSTEMS Description:a system of first-order differential or difference equations that govern the time evolution of the neuronal activations or membrane potentials Activation differential equations: (1) (2) in vector notation: (3) (4)

  15. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONAL DYNAMICAL SYSTEMS Neuronal State spaces So the state space of the entire neuronal dynamical system is: Augmentation:

  16. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONAL DYNAMICAL SYSTEMS Signal state spaces as hyper-cubes The signal state of field at time t: The signal state space: an n-dimensional hypercube The unit hypercube : or , The relationship between hyper-cubes and the fuzzy set : , subsets of correspond to the vertices of

  17. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS NEURONAL DYNAMICAL SYSTEMS Neuronal activations as short-term memory Short-term memory(STM) : activation Long-term memory(LTM) : synapse

  18. S k x o NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 1、Liner Function S(x) = cx + k , c>0

  19. S r -θ θ x -r NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 2. Ramp Function r if x≥θ S(x)= cx if |x|<θ -r if x≤-θ r>0, r is a constant.

  20. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 3、threshold linear signal function: a special Ramp Function Another form:

  21. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 4、logistic signal function: Where c>0. So the logistic signal function is monotone increasing.

  22. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 5、threshold signal function: Where T is an arbitrary real-valued threshold,and k indicates the discrete time step.

  23. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 6、hyperbolic-tangent signal function: Another form:

  24. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 7、threshold exponential signal function: When ,

  25. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 8、exponential-distribution signal function: When ,

  26. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION) 9、the family of ratio-polynomial signal function: An example For ,

  27. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION)

  28. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS SIGNAL FUNCTION (ACTIVATION FUNCTION)

  29. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS PULSE-CODED SIGNAL FUNCTION Definition: (5) (6) where (7)

  30. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS PULSE-CODED SIGNAL FUNCTION Pulse-coded signals take values in the unit interval [0,1]. Proof: when when

  31. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS PULSE-CODED SIGNAL FUNCTION Velocity-difference property of pulse-coded signals The first-order linear inhomogenous differential equation: (8) The solution to this differential equation: (9) (5) A simple form for the signal velocity: (10) (11)

  32. NEURONAL DYNAMICS Ⅰ: ACTIVATIONS AND SIGNALS PULSE-CODED SIGNAL FUNCTION (10) The central result of pulse-coded signal functions: The instantaneous signal-velocity equals the current pulse minus the current expected pulse frequency. ------------- the velocity-difference property of pulse-coded signal functions

More Related