. 2015 Jun 5;9:87. doi: 10.3389/fnsys.2015.00087

Table 2.

Model equations for learning and recall processes.

#	Equation	Description
RECALL
R1	$\begin{array}{l} {\hat{a}}_{j}^{(e)} = H (\sum_{i = 1}^{n} w_{i j}^{(e)} a_{i}^{(e)} - \sum_{i = 1}^{n} a_{i}^{(e)}) \\ H (x) = {\begin{array}{l} 1, if x \geq 0 \\ 0, if x < 0 \end{array} \end{array}$	The exteroceptive autoassociative network is presented a retrieval cue, a^(e) ∈ {0, 1}ⁿ, from the entorhinal cortex (EC). The retrieval activity pattern $\hat{a}$ ^(e) ∈ {0, 1}ⁿ is obtained by summing dendritic input from the recurrent connections for each cell, j, and applying a firing threshold $θ = \sum_{i}^{n} {\tilde{a}}_{i}^{(e)}$ .
R2	$\begin{array}{l} y_{l}^{(v)} = {\hat{y}}_{l}^{(v)} \land {\bar{I}}_{l} \\ {\hat{y}}_{l}^{(v)} = H (\sum_{j = 1}^{n} w_{j l}^{(e - v)} {\hat{a}}_{j}^{(e)} - \sum_{j}^{n} {\hat{a}}_{j}^{(e)}) \\ I_{l} = \lor_{i = 1}^{p \times m} I_{i l} {\hat{y}}_{i}^{(v)} \end{array}$	The output of the exteroceptive autoassociative network, $\hat{a}$ ^(e), drives retrieval in the heteroassociative network. An intermediate valence cell, l, can fire (y^(v)_l = 1) only if the dendritic sum of its excitatory inputs exceeds the specified threshold ( $\hat{y}$ ^(v)_l = 1) and if it does not receive inhibitory inputs from other valence cells that have already fired (Ī_j = 1 or I_j = 0).
R3	$\begin{array}{l} {\hat{a}}_{j}^{(i)} = H (\sum_{i = 1}^{m} w_{i j}^{(i)} {\tilde{a}}_{i}^{(i)} - \sum_{i = 1}^{m} {\tilde{a}}_{i}^{(i)}) \\ {\tilde{a}}_{k}^{(i)} = \lor_{i = 1}^{p \times m} w_{i k}^{(v - i)} y_{i}^{(v)} \end{array}$	The activity of the intermediate valence cells serves as input to the interoceptive autoassociative network through prewired connections, w^{(v − i)}_ik. The initial input is denoted by $\tilde{a}$ ⁽ⁱ⁾ and $\hat{a}$ ⁽ⁱ⁾ specifies the retrieved output, via recurrent connections, w⁽ⁱ⁾_ij.
LEARNING
L1	$w_{i j}^{(e)} (t + 1) = w_{i j}^{(e)} (t) \lor (a_{i}^{(e)} \land a_{j}^{(e)})$	The activity state of n cells in the exteroceptive autoassociative network is determined only by afferent inputs from the EC, a^(e) ∈ {0, 1}ⁿ. The recurrent synaptic weight w^(e)_ij between two cells i and j is clipped at 1 if both cells are active; otherwise, no change occurs. Initially, w^(e)_ij(0) = 0.
L2	$w_{i j}^{(i)} (t + 1) = w_{i j}^{(i)} (t) \lor (a_{i}^{(i)} \land a_{j}^{(i)})$	a⁽ⁱ⁾ ∈ {0, 1}^m specifies interoceptive, valence, activity pattern from the EC and w⁽ⁱ⁾_ij specifies synaptic weights of the interoceptive autoassociative network. Initially, w⁽ⁱ⁾_ij(0) = 0.
L3	$\begin{array}{l} w_{i j}^{(e - v)} (t + 1) = w_{i j}^{(e - v)} (t) \lor (a_{i}^{(e)} \land h_{j}^{(v)}) \\ h_{j}^{(v)} = x_{j}^{(v)} \land {\bar{I}}_{j} \\ x^{(v)} = (m_{1} a^{(i)}, m_{2} a^{(i)}, \dots, m_{p} a^{(i)}) \\ y^{(v)} = (g_{1}^{(v)}, g_{2}^{(v)}, \dots, g_{p}^{(v)}) \end{array}$	The intermediate valence cells are organized into p groups of m cells. As long as no inhibition is exerted on the primary group, its cells can be activated directly by interoceptive inputs from the EC (m₁ = 1). The interoceptive input, a⁽ⁱ⁾, on the remaining p − 1 groups of valence cells is gated by being multiplied by m_k signals that coincide with the detection of interference (Ξ_{k − 1} = 1). The interference condition verifies when the distance (HD) between the retrieval activity, y^(v)_j, across the valence cells of a group k (g^(v)_k) and the interoceptive activity from the EC (a^(v)) exceeds a threshold v. Initially, w^{(e − i)}_ij(0) = 0.
	$\begin{array}{l} m_{1} = 1, m_{k \in [2, p]} = {\begin{array}{l} 1, if Ξ_{k - 1} = 1 \\ 0, otherwise \end{array} \\ Ξ_{k} = \sum_{i = 1}^{m} g_{k_{i}}^{(v)} > 0 \land HD (g_{k}^{(v)}, a^{(v)}) > v \end{array}$