# Brain-State-in- a Box Network

## Brain-State-in- a Box Network

• The brain-State-in-a-Box (BSB) neural network refers to an easy nonlinear auto-associative neural network. It had been 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. A possible function of the BSB network is to spot a pattern from a given noisy version. The BSB network also can 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 equation,

Brain-State-in- a Box Network

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.

## 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've a brain that's placed during 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:

Brain-State-in- a Box Network

• 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.

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