Clustering and Anomaly Detection

Neuronal Nets, Rinzel

Josh Moller-Mara

NYU
j.mollermara@nyu.edu

What's clustering?

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

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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)

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Connectivity

Determine how cells are connected to each other.
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}

Connectivity Kernel

updateMiMe(1, 2) $m_I = 1, m_E = 2$

updateMiMe(1, 3) $m_I = 1, m_E = 3$

updateJScope(2, 3, 1, 1) $m_I = 2, m_E = 3$

updateJScope(1, 2, 1, 3) $m_I = 1, m_E = 2, j_I = 1, j_E = 3$

updateJScope(1, 5, 1, 3) $m_I = 1, m_E = 5, j_I = 1, j_E = 3$

updateJScope(1, 11, 2, 6) $m_I = 1, m_E = 11, j_I = 2, j_E = 6$

updateJScopeParam() $m_I$: $m_E$:
$jI$: $jE$:

Dynamics

We have two variables, $s$ which represents synaptic activation and $r$ which represents firing. \begin{align} \tau \frac{\partial}{\partial t}s(\theta, t) &= -s(\theta, t) + r(\theta, t)\\ r(\theta, t) &= \Phi \left[ \frac{1}{\pi} \int_{-\pi/2}^{\pi/2} J(\theta - \theta')s(\theta', t) d\theta' + I(\theta, t)\right] \end{align}

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

Current-to-Rate Transfer Function

updateSigmoid(2, 0) $\beta = 2, x_0 = 0$

updateSigmoid(6, 0) $\beta = 6, x_0 = 0$

updateSigmoid(6, 1) $\beta = 6, x_0 = 1$

updateSigmoidParam() $\beta$: $x_0$:

Stimulus to current

updateStimulus(40, 20) $m_s = 40, I_s = 20$

updateStimulus(40, 10) $m_s = 40, I_s = 10$

updateStimulus(1, 10) $m_s = 1, I_s = 10$

updateStimulusParam() $m_s$: $I_s$:

Other parameters

Max Time:
Time Step:
Number of time steps:

Delta Theta:
Number of cells:

$\tau$:

Show R:
Anomaly Detector:

Model

Input

autoUpdate(50) stopUpdate() Run model

Stop

clear() Clear

Anomaly detection

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

Changes to make an anomaly detector

I add a new variable $x(\theta, t)$, which is the firing rate of our anomaly detecting neurons.

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