HD2022

Victor Navarro

The mathematics behind HeiDI

The HeiDI model has four major components: 1) the acquisition of reciprocal associations between stimuli, 2) the pooling of those associations into stimulus activations, 3) the distribution of those activations into stimulus-specific response units, and 4) the generation of responses.

1 - Acquiring reciprocal associations

Whenever a trial is given, HeiDI learns associations among stimuli. The association between two stimuli, $i$ and $j$ is denoted via $v_{i,j}$. The association $v_{i,j}$ represents a directional expectation: the expectation of $j$ after being presented with $i$. Furthermore, its value represents the nature of the effect that $i$ has over the representation of $j$. If positive, the presentation of $i$ “excites” the representation of $j$. If negative, the presentation of $i$ “inhibits” the representation of $j$.

HeiDI not only learns “forward” associations between stimuli, but also their reciprocal, or “backward” associations. Thus, if organisms are presented with $i \rightarrow j$, organisms not only learn about $v_{i,j}$, but also about $v_{j, i}$, or the expectation of receiving $i$ after being presented with $j$. Note that, for the sake of brevity, the learning equations below are only specified for forward associations.

1.1 - The stimulus expectation rule

HeiDI generates expectations about stimuli. The expectation of stimulus $j$ ($e_j$) is expressed as

$$\tag{Eq. 1} e_j = \sum_{k}^{K}x_kv_{k,j}$$

where $K$ is the set containing all stimuli in the experiment, and $x_k$ is a quantity denoting the presence or absence of stimulus $k$ (1 or 0, respectively)¹.

1.2 - Learning rule

HeiDI learns the appropriate expectations via error-correction mechanisms. After trial $t$, the association between stimuli $i$ and $j$ is expressed as

$$\tag{Eq. 2} v_{i,j, t} = v_{i,j, t-1} + \Delta v_{i,j, t}$$

where $v_{j,i, t-1}$ is the forward association between $i$ and $j$ on trial $t-1$, and $\Delta v_{i,j, t}$ is the change in that association as a result of trial $t$. That delta term uses a pooled error term and is expressed as

$$\tag{Eq. 3} \Delta v_{i,j} = x_i\alpha_i(x_jc\alpha_j - e_j)$$ where $\alpha_i$ and $\alpha_j$ are parameters representing the salience of stimuli $i$ and $j$, respectively ($0 \le \alpha \le 1$), $c$ is a scaling constant ($c = 1$). Note that the term denoting the trial, $t$ has been omitted here for simplicity.

2 - Pooling the strength of associations

HeiDI pools its stimulus associations to activate stimulus-specific representations. The activation of the representation for stimulus $j$, $a_j$, is defined as:

$$\tag{Eq. 4} a_{j,M} = o_{j,M} + h_{j,M}$$

where $o_{j,M}$ denotes the combined associative strength towards stimulus $j$ in presence of stimuli $M$, and $h_{j,M}$ denotes the chained associative strength towards stimulus $j$ in presence of stimuli $M$.

2.1 - Combined associative strength

The quantity $o_{j,M}$ is the result of combining the associative strength of forward and backward associations to and from stimulus $j$ as

$$\tag{Eq. 5} o_{j,M} = \sum_{m \neq j}^{M}v_{m,j} + \left(\frac{\sum_{m \neq j}^{M}v_{m,j} \sum_{m \neq j}^{M}v_{j,m}}{c}\right)$$

where each of the sums above run over all stimuli $M$ presented in the trial, different from stimulus $j$.² The left-hand term describes how the forward associations from stimuli $M$ to $j$ affect the representation of $j$, whereas the right-hand term describes how the backward associations that $j$ has with stimuli $M$ affect its representation (although these are modulated by the forward associations themselves).

2.2 - Chained associative strength

The quantity $h_{j,M}$ captures the indirect associative strength that the stimuli $M$ have with $j$, via absent stimuli. As such, $h_{j,M}$ is defined as

$$\tag{Eq. 6a} h_{j,M} = \sum_{m \neq j}^{M} \sum_{n}^{N}\frac{v_{m,n}o_{j,n}}{c}$$

where N are the stimuli not presented on the trial (i.e., K-M). Note the re-use of $o$, the quantity defined in Eq. 5. This equation allows absent stimuli $N$ to influence the representation of stimulus $j$, as long as they have an association with present stimuli $M$.

In Honey and Dwyer (2022), the authors specify a similarity-based mechanism that modulates the effect of associative chains according to the similarity of the salience of nominal and retrieved stimuli³. As such, Eq. 6a is expanded as:

$$\tag{Eq. 6b} h_{j,M} = \sum_{m \neq j}^{M} \sum_{n}^{N}S(\alpha_{n}, \alpha'_n)\frac{v_{m,n}o_{j,n}}{c}$$

where $S$ is a similarity function that takes the nominal salience of stimulus n, $\alpha_n$ (as perceived when $n$ is presented on a trial) and its retrieved salience, $\alpha'_n$ (as perceived when $n$ is retrieved via other stimuli M, see ahead). This function is defined as:

$$\tag{Eq. 7} S(\alpha_n, \alpha'_n) = \frac{\alpha_n}{\alpha_n + |\alpha_n-\alpha'_n|} \times \frac{\alpha'_n}{\alpha'_n+ |\alpha_n-\alpha'_n|}$$

Notably, whenever there is more than one nominal salience for a given stimulus, then $\alpha_n$ is the arithmetic mean among all nominal values (see “heidi_similarity” vignette).

3 - Distributing strength into stimulus-specific response units

HeiDI then distributes the pooled stimulus-specific strength among all $K$ stimuli, according to their relative salience. The activation of response unit $j$, $R_j$ is expressed as

$$\tag{Eq. 8} R_{j,k} = \frac{\theta(j)}{\sum_{k}^{K}\theta(k)}a_{k,M}$$

where $j \in K$. As $K$ can include both present and absent stimuli, the $\theta$ function above depends on whether the stimulus $k$ is absent (i.e., $k \in N$) or not (i.e., $k \in M$), as:

$$\tag{Eq. 9} \theta(k) = \begin{cases} \left |\sum_{m}^{M}\left( v_{m,k}+\sum_{n \neq k}^{N}\frac{v_{m,n}v_{n,k}}{c}\right) \right|,& \text{if } k \in N\\ \alpha_k, & \text{otherwise} \end{cases}$$

Note that the quantity for absent stimuli is absolute, to prevent negative $\theta$ values due to inhibitory associations⁴. Also, note a summation term is used on the left-hand side of the expression for an absent stimulus. It implies that all the present stimuli $M$ contribute to the salience of stimulus $k$. Finally, note on the right-hand side of the same expression that the present stimuli contribute not only via the direct association each of them has with $k$, $v_{m,k}$ but also through associative chains with other absent stimuli (c.f., Eq. 6a).

4 - Generating responses

Finally, HeiDI responds. The response-generating mechanisms in HeiDI are currently underspecified. In its current version, HeiDI’s responses are the product of the activation of stimulus-specific response units and the connection that those units have with specific motor units. As such, the activation of motor unit $q$, $r_q$, is given by

$$\tag{Eq. 10} r_q = R_jw_{j,q}$$

where $w_{j,q}$ is a weight representing the association between stimulus-specific unit $j$ and motor unit $q$.