# 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) = x_{0},

Where,

**x(k) ∈ R ^{n }**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 : R ^{n }→ R^{n } 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: - 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.

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 ^{T}W_{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 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.**

## 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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