NEURAL NETWORKS AND FUZZY SYSTEMS

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NEURAL NETWORKS AND FUZZY SYSTEMS

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Chapter 2. Neural DynamicsActivation and Signals
?2.1 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)).
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Chapter 2. Neural DynamicsActivation and Signals
?2.1 Neurons as functions
The transduction description a sigmoidal or
S-shaped curve
e.g.1 the logistic signal function
(1)
(2)
Thus the logistic signal function is sigmoidal
and strictly increases for cgt0
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Chapter 2. Neural DynamicsActivation and Signals
?2.1 Neurons as functions
Fig2.1 s(x)x
If c?8,we get threshold signal function (dash
line), Which is piecewise differentiable
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Chapter 2. Neural DynamicsActivation and Signals
?2.2 Signal Monotonicity
In general,signal functions are monotone
nondecreasing Which means signal functions have
an upper bound or saturation value
case1differentiable case2piecewise-differenti
able
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Chapter 2. Neural DynamicsActivation and Signals
?2.2 Signal Monotonicity
An important exception Gaussian signal functions
(3)
(4)
We shall assume signal functions are monotone
nondecreasing unless stated otherwise
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Chapter 2. Neural DynamicsActivation and Signals
?2.2 Signal Monotonicity
Potential or radial basis function for
generalized Gaussian signal function
(5)
In which
(6)
(7)
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Chapter 2. Neural DynamicsActivation 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
networks computational richness and facilitates
noise suppression risks computational and
analytical intractability favors dynamical
instability.
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Chapter 2. Neural DynamicsActivation and Signals
?2.2 Signal Monotonicity
Signal and activation velocities dS/dt ,denoted
(8)
Signal velocities depend explicitly on action
velocities
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Chapter 2. Neural DynamicsActivation and Signals
?2.3 Biological activations and signals
The behave mechanism of the neuron
Fig2.Neuron anatomy
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Chapter 2. Neural DynamicsActivation and Signals
?2.3 Biological activations and signals
Competitive Neuronal Signal
Nonnegative signals often describe the
competitive status of neurons competing in a
laterally inhibitory neuron field
a. Binary signal functions b. Bipolar signal
functions
Eg2. Logical signal function
(9)
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Chapter 2. Neural DynamicsActivation and Signals
?2.4 Neuron Field
Neuron fielda topological group of neurons
zeroth-order topology and nonzeroth-order
topology Denotation Input field ,contains n
neurons Output field ,contains p
neurons Experiences or samples ,m vector
associations. The overall neural network
behaves as an adaptive filter,and sample data
changed network parameters.
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Chapter 2. Neural DynamicsActivation and Signals
?2.5 Neuronal Dynamical Systems
Descriptiona system of first-order differential
or difference equations that govern the time
evolution of the neuronal activations or
membrane potentials Activation differential
equations
(10)
(11)
in vector notation
(12)
(13)
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Chapter 2. Neural DynamicsActivation and Signals
?2.5 Neuronal Dynamical Systems
Neuronal State spaces
(14)
(15)
So the state space of the entire neuronal
dynamical system is
(16)
Augmentation
(17)
(18)
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Chapter 2. Neural DynamicsActivation and Signals
?2.5 Neuronal Dynamical Systems
Signal state spaces as hyper-cubes
The signal state of field at time t
(19)
The signal state space an n-dimensional
hypercube, brain states in a box The
relationship between hyper-cubes and the fuzzy
set The fit(fuzzy unit ) value
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Chapter 2. Neural DynamicsActivation and Signals
?2.5 Neuronal Dynamical Systems
Neuronal activations as short-term
memory Short-term memory(STM) Long-term
memory(LTM) Descartespineal glands Identify
neuronal activations with sense modalities
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
1?logistic signal function
(20)
Where cgt0.
(21)
So the logistic signal function is monotone
increasing.
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
2?threshold signal function
(22)
Where T is an arbitrary real-valued threshold,and
k indicates the discrete time step.
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
3?hyperbolic-tangent signal function
(23)
(24)
Another form
(25)
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
4?threshold linear signal function
(26)
Another form
(27)
(28)
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
5?threshold exponential signal function
(29)
When ,
(30)
(31)
(32)
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
6?exponential-distribution signal function
(33)
When ,
(34)
While
(35)
The diminishing returns effect.
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Chapter 2. Neural DynamicsActivation and Signals
?2.6 Common Signal Functions
7?the family of ratio-polynomial signal
function An example
(36)
For ,
(37)
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Chapter 2. Neural DynamicsActivation and Signals
?2.7 Pulse-Coded Signal Functions Definition
(38)
(39)
where
(40)
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Chapter 2. Neural DynamicsActivation and Signals
?2.7 Pulse-Coded Signal Functions Pulse-coded
signals take values in the unit interval
0,1. Proof extreme case 1,when
(41)
extreme case 2,when
(42)
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Chapter 2. Neural DynamicsActivation and Signals
?2.7 Pulse-Coded Signal Functions Velocity-differe
nce property of pulse-coded signals The
first-order linear inhomogenous differential
equation
(43)
The solution to this differential equation
(44)
A simple form for the signal velocity
(45)
(46)
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Chapter 2. Neural DynamicsActivation and Signals
?2.7 Pulse-Coded Signal Functions
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