j.mollermara@nyu.edu

- Clustering, or grouping, is fundamental to sensory input
- It allows us to recognize objects instead of disparate stimuli
*e.g.*Identifying speaker by voice- Separating objects in visual space

General structure: Continuous attractor model

- In this case, it's a ring shape, but it doesn't necessarily need to be
- The input space is from $-\pi/2$ to $\pi/2$
- The number of cells is taken to the limit of infinity (but here I'll use a discrete number)

We'd like for nearby cells to excite each other and far away cells to inhibit each other.

\begin{align} J(\theta) &= J_E(\theta) - J_I(\theta)\\ &= j_E \frac{\exp\left(m_E \cos(2\theta)\right)}{I_0(m_E)} - j_I \frac{\exp\left(m_I \cos(2\theta)\right)}{I_0(m_I)} \end{align}

$jI$: $jE$:

\begin{align}
\Phi(x) &= \frac{1}{1 + exp(-\beta[x - x_0])}
\end{align}

Time Step:

Number of time steps:

Delta Theta:

Number of cells:

$\tau$:

Show R:

Anomaly Detector:

Input

S

R

Anomaly Detector

Stop

- Anomaly detection is basically detecting when something unexpected happens
- It's used for detecting fraud, hacking attempts, among other things
- We take a simple version of anomaly detection, where we're interested in new percepts/groups we haven't seen before.
- By modifying the clustering network, we can achieve a simple anomaly detector

\begin{align} x(\theta, t) &= \Phi(I(\theta, t)) * (1 - s(\theta, t)) \end{align}

However, this may not be biologically plausible.

- Anomaly detection time can be changed by modifying $\tau$
- Modifying kernel can determine cluster width and ability to form
- Threshold is modified by changing $x_0$