Brain-State-in- a Box Network

The brain-State-in-a-Box (BSB) neural network refers to a simple nonlinear auto-associative neural network. It was proposed by J.A. Anderson, J.W. Silverstein, S.A. Ritz, and R.S. Jones in 1997 as a memory model that depends on neurophysiological considerations. The BSB model gets its name from the way that the network trajectory is forced to locate in the hypercube H_n= [-1, 1]_n. The BSB model was principally used to model the effects and mechanisms found in psychology and cognitive science. A possible function of the BSB network is to identify a pattern from a given noisy version. The BSB network can also be used as a pattern identifier that utilizes a smooth proximity measure and generates stable decision boundaries.

The elements of the BSB neural network are described by the differential equation,

x(t + 1) = g(x(k) + αW x(k)),

With an initial condition x(0) = x₀,

Where,

x(k) ∈ Rⁿ is the condition of the BSB neural network at time t.

α > 0 is a step size.

W ∈ R^n*n is an asymmetric weight matrix.

g : Rⁿ→ Rⁿ n is an activation function defined as a standard linear saturation function.

Some significant points about the BSB Network-

• BSB is an entirely associated network with the maximum number of nodes relying upon the dimension n of the input space.
• Neurons accept values between -1 to +1.
• All the neurons are updated at the same time.

BSB(brain-state-in-abox) Model:

The "brain-state-in-abox" sounds like we have a brain that is placed in a box without a body. The model is defined as follows:

Let us consider w be asymmetric weight matrix whose largest eigenvalues have positive and real components. Further, w is must be positive semi-definite.

x^TW_x>= 0 for all value of x

lets x(0) shows the initial state vector.

The BSB algorithm can be defined by these pair of equation:

P(n) = x(n) + ɳ W_x(n) ,

X(n+1) = f (p(n)).

We can say that the updating rule of the "brain state" x (a vector)

X → f (x + ɳ W_x)

Where,

ɳ = It shows a small constant called the feedback factor.

f = It is a linear function of the form

f(x) = +1       if x > 1 ;

f(x) = x       if -1 < x < -1;

f(x) = -1       if x < -1.

If the W is selecting with the given property (positive value of the largest eigenvalues), the impact of the algorithm is to drive the network for components of x to binary values +1 or -1 for each value of neuron. We can see it as networking from continuous inputs x(0) to discrete binary outputs. We get the final states that is in the form (-1,+1,-1,+1,-1,+1, ..., +1). It represents an edge of a cube in an N-dimensional space of liner size, centered at the origin. It is the box of the brain-state-in a box. The dynamics are like that the state shifts to the side of the box and then drives to the edge of the box.

The energy function of BSB

The energy function is also known as the Lyapunov function. The following equation gives the energy function of BSB:

E = -(ɳ/2) K_ij w_ij x_i x_j = -(ɳ/2) x^T W x

The equation mentioned above shows that the BSB dynamics minimize energy. It produces more general conditions that exist to choose when an energy function exists.