From c98f0e1bbf008320fa980fe36f61ff7a4e8baa04 Mon Sep 17 00:00:00 2001 From: Jim Martens Date: Wed, 20 Jun 2018 12:25:29 +0200 Subject: [PATCH] [NN] Added equation references Signed-off-by: Jim Martens --- neural-networks/seminarpaper.tex | 46 ++++++++++++++++---------------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/neural-networks/seminarpaper.tex b/neural-networks/seminarpaper.tex index 1bcb7ed..7b9e688 100644 --- a/neural-networks/seminarpaper.tex +++ b/neural-networks/seminarpaper.tex @@ -383,33 +383,33 @@ change probability \(p_i^w\) at time \(t\) is the product of the intrinsic weigh change probability \(W_i\) and the concentration of the neuromodulator the synapse is sensitive to \(c(t, x_i, y_i)\) at its location \((x_i, y_i)\). Additionally the maximum neuromodulator sensitivity \(M_i\) is the ceiling for the second part -of that product. This means there is a maximum weight change probability for each -synapse. Weight changes can happen at any time step. Therefore the intrinsic weight -change probability has to be very small. Should a weight change occur a new weight -\(w_i\) is chosen randomly from the interval \([W_i^{min}, W_i^{max}]\). +of that product \eqref{eq:weightchangeprob}. This means there is a maximum weight +change probability for each synapse. Weight changes can happen at any time step. +Therefore the intrinsic weight change probability has to be very small. Should a +weight change occur a new weight \(w_i\) is chosen randomly from the interval +\([W_i^{min}, W_i^{max}]\). The weight change probability \(p_i^w\) tells the network when to learn and leaves room for variation as it is a probability and not a binary learn/do not learn situation. Within this example this probability is the so called second environmental feedback loop. - -\[ +\begin{equation}\label{eq:weightchangeprob} p_i^w = min(M_i, c(t, x_i, y_i)) \cdot W_i,\; 0 < W_i \lll 1 -\] +\end{equation} Moreover a synapse can disable or enable itself. The actual disable/enable probability \(p_i^d\) is the product of the intrinsic value \(D_i\) saved as -parameter and the neuromodulator concentration \(c(t, x_i, y_i)\). The concentration -is again ceiled by the maximum sensitivity limit \(M_i\) given as parameter. -This means there is a maximum disable/enable probability as well. The intrinsic -enable/disable probability must be smaller than the intrinsic weight change probability. -A disabled synapse is treated as having weight 0 but the actual value is stored -so that it can be restored when the synapse is enabled again. +parameter and the neuromodulator concentration \(c(t, x_i, y_i)\) \eqref{eq:enableprob}. +The concentration is again ceiled by the maximum sensitivity limit \(M_i\) given +as parameter. This means there is a maximum disable/enable probability as well. +The intrinsic enable/disable probability must be smaller than the intrinsic weight +change probability. A disabled synapse is treated as having weight 0 but the actual +value is stored so that it can be restored when the synapse is enabled again. -\[ +\begin{equation}\label{eq:enableprob} p_i^d = min(M_i, c(t, x_i, y_i) \cdot D_i,\; 0 \leq D_i < W_i -\] +\end{equation} Given a so called neural network structure or substrate this makes it easier to find different network topologies (structure and weights combined). @@ -421,14 +421,14 @@ The modulated gaussian walk is introduced by Toutounji and Pasemann. The key dif start with the parameters. There is no maximum sensitivity for the neuromodulator concentration. When a weight change occurs the new weight is not chosen randomly but rather the difference to be added to the current weight is sampled from a -normal distribution with a mean of zero and \(\sigma^2\)-variance. The sampled -value could be infinitely large and hence the new weight outside of the given -bounds for it. Therefore the value is sampled until the sum of the +normal distribution with a mean of zero and \(\sigma^2\)-variance \eqref{eq:gausswalk}. +The sampled value could be infinitely large and hence the new weight outside of +the given bounds for it. Therefore the value is sampled until the sum of the current weight and the sampled value are within the interval \([W_i^{min}, W_i^{max}]\). -\[ +\begin{equation}\label{eq:gausswalk} w_i (t + 1) = w_i (t) + \Delta w_i \;\text{where}\; \Delta w_i \sim \mathcal{N}(0, \sigma^2) -\] +\end{equation} Toutounji and Pasemann implemented a mechanism for disabling synapses in the modulated gaussian walk as well but did not make use of it later and @@ -463,11 +463,11 @@ it when to learn. How does the actual learning happen? The weight change between two neurons is dependent on the activation of both neurons, the learning rate and the concentration -of neuromodulators. In short Hebbian learning is employed. +of neuromodulators \eqref{eq:hebbian}. In short Hebbian learning is employed. -\[ +\begin{equation}\label{eq:hebbian} \Delta w_{ij} = \eta \cdot m_i \cdot a_i \cdot a_j -\] +\end{equation} This explanation should suffice for the general understanding of their method. The neurons within the vicinity of these sources only update their weights