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# 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) = 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.

## 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 ^{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 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.