NEURONAL DYNAMICS 2: ACTIVATION MODELS

NEURONAL DYNAMICS 2: ACTIVATION MODELS

Chapter 3. Neuronal Dynamics 2 :Activation Models
3.1 Neuronal dynamical system Neuronal activations change with time. The way they change depends on the dynamical equations as following: (3-1) (3-2)

3.1 ADDITIVE NEURONAL DYNAMICS
first-order passive decay model In the absence of external or neuronal stimuli, the simplest activation dynamics model is: (3-3) (3-4)

3.1 ADDITIVE NEURONAL DYNAMICS
since for any finite initial condition The membrane potential decays exponentially quickly to its zero potential.

Passive Membrane Decay
Passive-decay rate scales the rate to the membrane’s resting potential. solution : Passive-decay rate measures: the cell membrane’s resistance or “friction” to current flow.

property Pay attention to property The larger the passive-decay rate,the faster the decay--the less the resistance to current flow.

Membrane Time Constants
The membrane time constant scales the time variable of the activation dynamical system. The multiplicative constant model: （3－8）

Solution and property solution property The smaller the capacitance ,the faster things change As the membrane capacitance increases toward positive infinity,membrane fluctuation slows to stop.

Membrane Resting Potentials
Definition Define resting Potential as the activation value to which the membrane potential equilibrates in the absence of external or neuronal inputs: (3-11) Solutions (3-12)

Note The capacitance appear in the index of the solution, it is called time-scaling capacitance. It does not affect the asymptotic or steady-state solution and does not depend on the finite initial condition.

Additive External Input
Add input Apply a relatively constant numeral input to a neuron. (3-13) solution (3-14)

Meaning of the input Input can represent the magnitude of directly experiment sensory information or directly apply control information. The input changes slowly,and can be assumed constant value.

3.2 ADDITIVE NEURONAL FEEDBACK
Neurons do not compute alone. Neuron modify their state activations with external input and with the feedback from one another. This feedback takes the form of path-weighted signals from synaptically connected neurons.

Synaptic Connection Matrices
n neurons in field p neurons in field The ith neuron axon in a synapse jth neurons in is constant,can be positive,negative or zero.

Meaning of connection matrix
The synaptic matrix or connection matrix M is an n-by-p matrix of real number whose entries are the synaptic efficacies the ijth synapse is excitatory if inhibitory if The matrix M describes the forward projections from neuron field to neuron field The matrix N describes the backward projections from neuron field to neuron field

Bidirectional and Unidirectional connection Topologies
Bidirectional networks M and N have the same or approximately the same structure. Unidirectional network A neuron field synaptically intraconnects to itself.M nxn. BAM M is symmetric, the unidirectional network is BAM

Augmented field and augmented matrix
M connects to ,N connects to then the augmented field intraconnects to itself by the square block matrix B

Augmented field and augmented matrix
In the BAM case,when then hence a BAM symmetries an arbitrary rectangular matrix M. In the general case, P is n-by-n matrix. Q is p-by-p matrix. If and only if, the neurons in are symmetrically intraconnected

3.3 ADDITIVE ACTIVATION MODELS
Define additive activation model n+p coupled first-order differential equations defines the additive activation model （3－15）（3－16）

additive activation model define
The additive autoassociative model correspond to a system of n coupled first-order differential equations （3－17）

additive activation model define
A special case of the additive autoassociative model (3-18) (3-19) where is （3-20） measures the cytoplasmic resistance between neurons i and j.

continuous additive bidirectional associative memories
Hopfield circuit and continuous additive bidirectional associative memories Hopfield circuit arises from if each neuron has a strictly increasing signal function and if the synaptic connection matrix is symmetric (3-21) continuous additive bidirectional associative memories (3-22) (3-23)

3.4 ADDITIVE BIVALENT FEEDBACK
Discrete additive activation models correspond to neurons with threshold signal function The neurons can assume only two value: ON and OFF. ON represents the signal value +1. OFF represents 0 or –1. Bivalent models can represent asynchronous and stochastic behavior.

BAM-bidirectional associative memory
Bivalent Additive BAM BAM-bidirectional associative memory Define a discrete additive BAM with threshold signal functions, arbitrary thresholds and inputs,an arbitrary but constant synaptic connection matrix M,and discrete time steps k. (3-24) (3-25)

Threshold binary signal functions
Bivalent Additive BAM Threshold binary signal functions (3-26) (3-27) For arbitrary real-value thresholds for neurons for neurons

A example for BAM model Example A 4-by-3 matrix M represents the forward synaptic projections from to A 3-by-4 matrix MT represents the backward synaptic projections from to

Suppose at initial time k all the neurons in are ON.
A example for BAM model Suppose at initial time k all the neurons in are ON. So the signal state vector at time k corresponds to Input Suppose

first:at time k+1 through synchronous operation,the result is:
A example for BAM model first:at time k+1 through synchronous operation,the result is: next:at time k+1 ,these signals pass “forward” through the filter M to affect the activations of the neurons. The three neurons compute three dot products,or correlations. The signal state vector multiplies each of the three columns of M.

A example for BAM model the result is: synchronously compute the new signal state vector :

A example for BAM model the signal vector passes “backward” through the synaptic filter at time k+2: synchronously compute the new signal state vector :

A example for BAM model since then conclusion These same two signal state vectors will pass back and forth in bidirectional equilibrium forever-or until new inputs perturb the system out of equilibrium.

A example for BAM model asynchronous state changes may lead to different bidirectional equilibrium keep the first neurons ON,only update the second and third neurons. At k,all neurons are ON. new signal state vector at time k+1 equals:

A example for BAM model new activation state vector equals: synchronously thresholds passing this vector forward to gives

A example for BAM model similarly, for any asynchronous state change policy we apply to the neurons the system has reached a new equilibrium,the binary pair represents a fixed point of the system.

conclusion conclusion Different subset asynchronous state change policies applied to the same data need not product the same fixed-point equilibrium. They tend to produce the same equilibria. All BAM state changes lead to fixed-point stability.

Bidirectional Stability
definition A BAM system is Bidirectional stable if all inputs converge to fixed-point equilibria. A denotes a binary n-vector in B denotes a binary p-vector in

Bidirectional Stability
Represent a BAM system equilibrates to bidirectional fixed point as

Lyapunov Functions Lyapunov Functions L maps system state variables to real numbers and decreases with time. In BAM case,L maps the Bivalent product space to real numbers. Suppose L is sufficiently differentiable to apply the chain rule: (3-28)

Lyapunov Functions The quadratic choice of L （3-29） Suppose the dynamical system describes the passive decay system. （3-30） The solution （3-31）

Lyapunov Functions The partial derivative of the quadratic L: （3-32）（3-33） or (3-34) (3-35) In either case (3-36) At equilibrium This occurs if and only if all velocities equal zero

conclusion A dynamical system is stable if some Lyapunov Functions L decreases along trajectories. A dynamical system is asymptotically stable if it strictly decreases along trajectories Monotonicity of a Lyapunov Function provides a sufficient not necessary condition for stability and asymptotic stability.

Linear system stability
For symmetric matrix A and square matrix B,the quadratic form behaves as a strictly decreasing Lyapunov function for any linear dynamical system if and only if the matrix is negative definite.

The relations between convergence rate and eigenvalue sign
A general theorem in dynamical system theory relates convergence rate and eigenvalue sign: A nonlinear dynamical system converges exponetially quickly if its system Jacobian has eigenvalues with negative real parts. Locally such nonlinear system behave as linearly. (Jacobian matrix) A Lyapunov Function summarizes total system behavior. A Lyapunov Function often measures the energy of a physical sysem Represents system energy decrease with dynamical systems

Potential energy function represented by quadratic form
Consider a system of n variables and its potential-energy function E. Suppose the coordinate measures the displacement from equilibrium of ith unit.The energy depends on only coordinate ,so since E is a physical quantity,we assume it is sufficiently smooth to permit a multivariable Taylor-series expansion about the origin:

Potential energy function represented by quadratic form
Where A is symmetric,since

The reason of (3-42)follows
First,we defined the origin as an equilibrium of zero potential energy;so Second,the origin is an equilibrium only if all first partial derivatives equal zero. Third,we can neglect higher-order terms for small displacement,since we assume the higher-order products are smaller than the quadratic products.

Conclusion: Bounded decreasing L funcs provide an intuitive way to describe global “computations” in nueral networks ad other dynamical system.

Bivalent BAM theorem The average signal energy L of the forward pass of the Signal state vector through M,and the backward pass Of the signal state vector through : since

Lower bound of Lyapunov function
The signal is Lyapunov function clearly bounded below. For binary or bipolar,the matrix coefficients define the attainable bound: The attainable upper bound is the negative of this expression.

Lyapunov function for the general BAM system
The signal-energy Lyapunov function for the general BAM system takes the form Inputs and and constant vectors of thresholds the attainable bound of this function is.

Bivalent BAM theorem Bivalent BAM theorem.every matrix is bidrectionally stable for synchronous or asynchronous state changes. Proof consider the signal state changes that occur from time k to time k+1,define the vectors of signal state changes as:

Bivalent BAM theorem define the individual state changes as: We assume at least one neuron changes state from k to time k+1. Any subset of neurons in a field can change state,but in only one field at a time. For binary threshold signal functions if a state change is nonzero,

Bivalent BAM theorem For bipolar threshold signal functions The “energy”change Differs from zero because of changes in field or in field

Bivalent BAM theorem

Bivalent BAM theorem Suppose Then This implies so the product is positive: Another case suppose

Bivalent BAM theorem This implies so the product is positive: So for every state change. Since L is bounded,L behaves as a Lyapunov function for the additive BAM dynamical system defined by before. Since the matrix M was arbitrary,every matrix is bidirectionally stable. The bivalent Bam theorem is proved.

Property of globally stable dynamical system

Two insights about the rate of convergence
First,the individual energies decrease nontrivially.the BAM system does not creep arbitrary slowly down the toward the nearest local minimum.the system takes definite hops into the basin of attraction of the fixed point. Second,a synchronous BAM tends to converge faster than an asynchronous BAM.In another word, asynchronous updating should take more iterations to converge.

Review 1.Neuronal Dynamical Systems
We describe the neuronal dynamical systems by first-order differential or difference equations that govern the time evolution of the neuronal activations or membrane potentials.

Review 4.Additive activation models Hopfield circuit:
Additive autoassociative model; Strictly increasing bounded signal function ; Synaptic connection matrix is symmetric

Review 5.Additive bivalent models Lyapunov Functions
Cannot find a lyapunov function,nothing follows; Can find a lyapunov function,stability holds.

Review A dynamics system is stable , if ; asymptotically stable, if .
Monotonicity of a lyapunov function is a sufficient not necessary condition for stability and asymptotic stability.

Review Bivalent BAM theorem.
Every matrix is bidirectionally stable for synchronous or asynchronous state changes. Synchronous:update an entire field of neurons at a time. Simple asynchronous:only one neuron makes a state-change decision. Subset asynchronous:one subset of neurons per field makes state-change decisions at a time.

Chapter 3. Neural Dynamics II:Activation Models
The most popular method for constructing M:the bipolar Hebbian or outer-product learning method binary vector associations: bipolar vector associations:

The binary outer-product law: The bipolar outer-product law: The Boolean outer-product law:

The weighted outer-product law: Where holds. In matrix notation: Where

※3.6.1 Optimal Linear Associative Memory Matrices Optimal linear associative memory matrices: The pseudo-inverse matrix of :

※3.6.1 Optimal Linear Associative Memory Matrices Optimal linear associative memory matrices: The pseudo-inverse matrix of : If x is a nonzero scalar: If x is a nonzero vector: If x is a zero scalar or zero vector : For a rectangular matrix , if exists:

※3.6.1 Optimal Linear Associative Memory Matrices Define the matrix Euclidean norm as Minimize the mean-squared error of forward recall,to find that satifies the relation

※3.6.1 Optimal Linear Associative Memory Matrices Suppose further that the inverse matrix exists. Then So the OLAM matrix correspond to

If the set of vector is orthonormal Then the OLAM matrix reduces to the classical linear associative memory(LAM) : For is orthonormal, the inverse of is

※3.6.2 Autoassociative OLAM Filtering Autoassociative OLAM systems behave as linear filters. In the autoassociative case the OLAM matrix encodes only the known signal vectors Then the OLAM matrix equation (3-78) reduces to M linearly “filters” input measurement x to the output vector by vector matrix multiplication:

※3.6.2 Autoassociative OLAM Filtering The OLAM matrix behaves as a projection operator[Sorenson,1980].Algebraically,this means the matrix M is idempotent: Since matrix multiplication is associative,pseudo-inverse property (3-80) implies idempotency of the autoassociative OLAM matrix M:

※3.6.2 Autoassociative OLAM Filtering Then (3-80) also implies that the additive dual matrix behaves as a projection operator: We can represent a projection matrix M as the mapping

※3.6.2 Autoassociative OLAM Filtering The Pythagorean theorem underlies projection operators. The known signal vectors span some unique linear subspace of L equals , the set of all linear combinations of the m known signal vectors. denotes the orthogonal complement space ,the set of all real n-vectors x orthogonal to every n-vector y in L.

※3.6.2 Autoassociative OLAM Filtering Operator projects onto L. The dual operator projects onto Projection Operator and uniquely decompose every vector x into a summed signal vector and a noise or novelty vector : x

※3.6.2 Autoassociative OLAM Filtering The unique additive decomposition obeys a generalized Pythagorean theorem: where defines the squared Euclidean or norm. Kohonen[1988] calls the novelty filter on

※3.6.2 Autoassociative OLAM Filtering Projection measures what we know about input x relative to stored signal vectors : for some constant vector The novelty vector measures what is maximally unknown or novel in the measured input signal x.

※3.6.2 Autoassociative OLAM Filtering Suppose we model a random measurement vector x as a random signal vector corrupted by an additive, independent random-noise vector : We can estimate the unknown signal as the OLAM-filtered output

※3.6.2 Autoassociative OLAM Filtering Kohonen[1988] has shown that if the multivariable noise distribution is radially symmetric, such as a multivariable Gaussian distribution,then the OLAM capacity m and pattern dimension n scale the variance of the random-variable estimator-error norm :

※3.6.2 Autoassociative OLAM Filtering 1.The autoassociative OLAM filter suppress noise if , when memory capacity does not exceed signal dimension. 2.The OLAM filter amplifies noise if , when capacity exceeds dimension.

※3.6.3 BAM Correlation Encoding Example The above data-dependent encoding schemes add outer-product correlation matrices. The following example illustrates a complete nonlinear feedback neural network in action,with data deliberately encoded into the system dynamics.

※3.6.3 BAM Correlation Encoding Example Suppose the data consists of two unweighted binary associations and defined by the nonorthogonal binary signal vectors:

※3.6.3 BAM Correlation Encoding Example These binary associations correspond to the two bipolar associations and defined by the bipol –ar signal vectors:

※3.6.3 BAM Correlation Encoding Example We compute the BAM memory matrix M by adding the bipol –ar correlation matrices and pointwise. The first correlation matrix equals

※3.6.3 BAM Correlation Encoding Example Observe that the i th row of the correlation matrix equals the bipolar vector multipled by the i th element of The j th column has the similar result. So equals

※3.6.3 BAM Correlation Encoding Example Adding these matrices pairwise gives M:

※3.6.3 BAM Correlation Encoding Example Suppose, first,we use binary state vectors.All update policies are synchronous.Suppose we present binary vector as input to the system—as the current signal state vector at Then applying the threshold law (3-26) synchronously gives

※3.6.3 BAM Correlation Encoding Example Passing through the backward filter , and applying the bipolar version of the threshold law(3-27),gives back : So is a fixed point of the BAM dynamical system. It has Lyapunov “energy” , which equals the backward value has the similar result:a fixed point with energy

※3.6.3 BAM Correlation Encoding Example So the two deliberately encoded fixed points reside in equally “deep” attractors. Hamming distance H equals distance counts the number of slots in which binary vectors and differ:

※3.6.3 BAM Correlation Encoding Example Consider for example the input , which differs from by 1 bit , or Then Fig3.2 shows that BAM can return original balance regardless of the noise. bipolar

※ Memory Capacity:Dimensionality Limits Capacity Synaptic connection matrices encode limited information. We sum more correlation matrices ,then holds more frequently. After a point,adding additional associations Does not significantly change the connection matrix. The system “forgets”some patterns. This limits the memory capacity.

※ Memory Capacity:Dimensionality Limits Capacity Grossberg’s sparse coding theorem [1976] says , for deterministic encoding ,that pattern dimensionality must exceed pattern number to prevent learning some patterns at the expense of forgetting others.

※3.6.5 The Hopfield Model The Hopfield model illustrates an autoassociative additive bivalent BAM operated serially with simple asynchronous state changes. Autoassociativity means the network topology reduces to only one field, ,of neurons: The synaptic connection matrix M symmetrically intraconnects the n neurons in field

※3.6.5 The Hopfield Model The autoassociative version of Equation (3-24) describes the additive neuronal activation dynamics: (3-87) for constant input , with threshold signal function (3-88)

※3.6.5 The Hopfield Model We precompute the Hebbian synaptic connection matrix M by summing bipolar outer-product(autocorrelation)matrices and zeroing the main diagonal: (3-89) where I denotes the n-by-n identity matrix . Zeroing the main diagonal tends to improve recall accuracy by helping the system transfer function behave less like the identity operator.

※3.7 Additive dynamics and the noise-saturation dilemma Grossberg’s Saturation Theorem Grossberg’s Saturation theorem states that additive activation models saturate for large inputs, but multiplicative models do not .

The stationary “reflectance pattern” confronts the system amid the background illumination The i th neuron receives input .Convex coefficient defines the “reflectance” : : the passive decay rate : the activation bound

Additive Grossberg model: We can solve the linear differential equation to yield For initial condition , as time increases the activation converges to its steady-state value: As

So the additive model saturates. Multiplicative activation model:

For initial condition ,the solution to this differential equation becomes As time increases, the neuron reaches steady state exponentially fast: (3-96) as

This proves the Grossberg saturation theorem: Additive models saturate ,multiplicative models do not.

In general the activation variable can assume negative values . Then the operating range equals for In the neurobiological literature the lower bound is usually smaller in magnitude than the upper bound : This leads to the slightly more general shunting activation model:

Setting the right-hand side of the above equation to zero, and we can get the equilibrium activation value: which reduces to (3-96) if C=0. the neuron generates nonnegative activations.

※3.8 General Neuronal Activations:Cohen-Grossberg and multiplicative models Consider the symmetric unidirectional or autoassociative case when , , and M is constant . Then a neural network possesses Cohen-Grossberg[1983] activation dynamics if its activation equations have the form (3-102) The nonnegative function represents an abstract amplification function.

Grossberg[1988]has also shown that (3-102) reduces to the additive brain-state-in-a-box model of Anderson[1977,1983] and the shunting masking-field model [Cohen,1987] upon appropriate change of variables.

If , , and constant , where and are positive constants , and input is constant or varies slowly relative to fluctuations in ,then (3-102) reduces to the Hopfield circuit[1984]: An autoassociative network has shunting or multiplicative activation dynamics when the amplification function is linear, and is nonlinear .

For instance , if , (self-excitation in lateral inhibition) , and then (3-104) describes the distance-dependent unidirectional shunting network :

Hodgkin-Huxley membrane equation: , and denote respectively passive(chloride ) , excitatory (sodium ) , and inhibitory (potassium ) saturation upper bounds .

At equilibrium, when the current equals zero ,the Hodgkin-Huxley model has the resting potential : Neglect chloride-based passive terms.This gives the resting potential of the shunting model as

BAM activations also possess Cohen-Grossberg dynamics, and their extensions: with corresponding Lyapunov function L , as we show in Chapter 6 :

谢谢大家！

NEURONAL DYNAMICS 2: ACTIVATION MODELS

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