# 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 Hn= [-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 ∈ Rn*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.

xTWx>= 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 Wx(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 Wx)

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) Kij wij xi xj = -(ɳ/2) xT 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.   