Timing using temporal context

Abstract

We present a memory model that explicitly constructs and stores the temporal information about when a stimulus was encountered in the past. The temporal information is constructed from a set of temporal context vectors adapted from the temporal context model (TCM). These vectors are leaky integrators that could be constructed from a population of persistently firing cells. An array of temporal context vectors with different decay rates calculates the Laplace transform of real time events. Simple bands of feedforward excitatory and inhibitory connections from these temporal context vectors enable another population of cells, timing cells. These timing cells approximately reconstruct the entire temporal history of past events. The temporal representation of events farther in the past is less accurate than for more recent events. This history-reconstruction procedure, which we refer to as timing from inverse Laplace transform (TILT), displays a scalar property with respect to the accuracy of reconstruction. When incorporated into a simple associative memory framework, we show that TILT predicts well-timed peak responses and the Weber law property, like that observed in interval timing tasks and classical conditioning experiments.

Figure 1

TCM architecture. The left panel shows two layers with nodes representing the stimulus layer and the context layer. The external input activates a single node in the stimulus layer. The operator C in turn activates the corresponding node in the context layer. But the other nodes in the context layer corresponding to the stimuli encountered in recent past are still active. This is represented by the shaded activity in different context nodes. The lighter the shading, the farther back in time the corresponding stimulus was experienced. At each moment the context activity is associated with the incoming stimulus and is stored in the matrix M. An example is illustrated in the right panel. The context activity gradually changes from t₁ to t₂ to t₃ as three stimuli are sequentially presented. The context activity prior to the experience of a stimulus gets stored in a row of M uniquely associated with that stimulus. Hence t₁ is stored in the tree row of M, t₂ is stored in the cat row and t₃ is stored in the pen row. The stored information in M can be accessed through the context activity at any point in the future. At the retrieval phase, if the context activity is given by t_cue, the rows of M which are similar to t_cue get strongly activated and the rows which are less similar to t_cue get less activated. In this example, we have chosen t_cue to be more similar to t₃, hence we see that the component of p vector corresponding to pen is stronger than the other components.

Figure 2

Temporal decay of continuous-time context activity. The top curve represents the stimulus presented thrice with different durations and intensities. The bottom curve represents the activity of the corresponding context node. We have taken ρ = 0.3. The dotted line intersects the curve representing the context activity at six points, indicating that the context activity at these points is the same despite different stimulus history preceding each point. The y axis has arbitrary units and the two curves are not drawn to scale.

Figure 3

Schematic representation of multiple context vectors stacked together. The different vectors are ordered by their ρ or s values with components corresponding to each stimulus lined up. All the nodes within each column of the t layer are activated by a specific f node. As an illustration, two columns of t are shaded in concordance with their corresponding f node.

Figure 4

Schematic description of the one to one mapping between the context layer t and the timing layer T. The activity in each t column is mapped on to the activity in the corresponding T column via the operator ${\mathbf{L}}_k^{-1}$ according to equation 6.

Figure 5

Scalar property of the reconstructed stimulus history. Four stimuli of duration 0.1 sec was presented at various moments in the recent past. The curves show the reconstructed stimulus history for these stimuli, with a fixed value of k = 12. The coefficient of variation, the mean divided by the standard deviation, of each of these curves is exactly the same. The qualitative features of the graph is the same for all k, but the coefficient of variation decreases for higher values of k. The table gives the coefficient of variation of these curves for each stimulus (columns) for different values of k (rows).

Figure 6

Reconstruction of complex stimulus history. A stimulus is presented twice in the recent past and the reconstructed history is plotted for k = 12 (left) and k = 4 (right). For the k = 12 case, the two peaks clearly occur at the appropriate positions, with the more recent stimulus being better represented. For the k = 4 case, the earlier peak is barely discernible.

Figure 7

Time dependent activity of various layers of the model. A stimulus is presented twice, and the activity of two cells in the corresponding t and T columns are shown as a function of time. Note that the activity of the T cells peak roughly at the appropriate delay after each stimulus presentation.

Figure 8

Neural Representation of the internal time. The left most panel shows a column of the context layer t, and the associated column in the timing layer T. The cells in these two columns are mapped in a one to one fashion. The activity of any cell in the T column not only depends on the activity of its counterpart in the t column, but also on the activity of k neighbors in the t column. This is a discretized approximation of ${\mathbf{L}}_k^{-1}$ from eq. 6. The right panel gives a pictorial representation of the connectivity between a cell in the T column and its k near neighbors in the t column. The contribution from the neighbors alternate between excitation and inhibition in either directions. The points above x-axis are excitations and the points below the x-axis are inhibitions. The tick marks on x-axis denote the position of the neighboring cells on either side. The dotted curve that forms an envelope simply helps to illustrate that the magnitude of the contribution falls off with the distance to the neighbor. With k = 2, we see an off-center-on-sorround connectivity. With k = 4, we see a mexican-hat like connectivity, and k = 12 shows a more elaborate band of connectivity.

Figure 9

Timing mechanism and memory. The external input at any moment activates a unique node in the stimulus layer f. Corresponding to each stimulus node is a column of cells in the context layer t that get activated via C according to Eq. 4. Each cell in this column has a distinct decay rate spanning 0 < ρ < 1. Each column of the t layer is mapped on to the corresponding column in the T layer via ${\mathbf{L}}_k^{-1}$ described in Figure 8. The T layer activity at each moment is associated in a Hebbian fashion with the f layer activity and these associations are stored in M. After sufficient training with sequential external inputs, the associations in M can grow significantly strong and the T layer activity at any moment can induce activity in the f layer through M. This internally generated activity in the stimulus layer is interpreted as the prediction p for the next moment.

Figure 10

Timing in goldfish. During training, the US (shock) was presented 5 sec (top panel) and 15 sec (bottom panel) after the onset of the CS (light). The rate of CR is plotted in the left panel as a function of the time after presentation of the CS in the absence of the US. This figure is reproduced from Drew et al (2005). The different curves represent different number of learning trials. Notice that the response gets stronger with learning trials. The right panel shows the probability of CR generated from simulations of the model. In these simulations, for simiplicity, only the onset of CS is encoded into the context, not the entire CS. The parameters used in this simulation are k = 4, θ = 0.1 and φ = 1.