Timing in the absence of clocks: encoding time in neural network states

Abstract

Decisions based on the timing of sensory events are fundamental to sensory processing. However, the mechanisms by which the brain measures time over ranges of milliseconds to seconds remain unclear. The dominant model of temporal processing proposes that an oscillator emits events that are integrated to provide a linear metric of time. We examine an alternate model in which cortical networks are inherently able to tell time as a result of time-dependent changes in network state. Using computer simulations we show that within this framework, there is no linear metric of time, and that a given interval is encoded in the context of preceding events. Human psychophysical studies were used to examine the predictions of the model. Our results provide theoretical and experimental evidence that, for short intervals, there is no linear metric of time, and that time may be encoded in the high-dimensional state of local neural networks.

Figure 1. State-dependent network simulation

(A) Voltage plot of a subset of neurons in the network. Each line represents the voltage of a single neuron in response to two identical events separated by 100 ms. The first 100 lines represent 100 Ex units (out of 400), and the remaining lines represent 25 Inh units (out of 100). Each input produces a depolarization across all neurons in the network, followed by inhibition. While most units exhibit subthreshold activity, some spike (white pixels) to both inputs, or exclusively to the first or second. The Ex units are sorted according to their probability of firing to the first (top) or second (bottom) pulse. This selectivity to the first or second event arises because the difference in network state at t=0 and t=100 ms. (B) Trajectory of the three principal components of the network in response to a single pulse. There is an abrupt and rapidly evolving response beginning at t=0, followed by a slower trajectory. The fast response is due to the depolarization of a large number of units, while the slower change reflects the short-term synaptic dynamics and slow IPSPs. The speed of the trajectory in state space can be visualized by the rate of change of the color code and by the distance between the 25 ms marker spheres. Because synaptic properties cannot be rapidly 'reset', the network cannot return to its initial state (arrow) before the arrival of a second event. (C) Trajectory in response to a 100 ms interval. Note that the same stimulus produces a different fast response to the second event. To allow a direct comparison, the principal components from B were used to transform the state data in C.

Figure 2. Encoding of Temporal Patterns

(A) Information per neuron. The blue trace displays the mutual information that each Ex unit provides for the discrimination of a 100 versus 200 ms interval (sorted). The red line shows the information for the same intervals preceded by a 150 ms interval, that is discrimination of the pattern [150;100] versus [150;200]. While individual neurons contain significant information for both stimuli, a different population of neurons encodes each one. (B) Discrimination of all four stimuli. All Ex units were connected to four output neurons trained to recognize the network activity produced by the last pulse of all four stimuli. Average responses were calculated from six independent (different random number generator seeds) simulations. Note that a mutual information measure based on total spike count to each stimulus, as in panel A, would introduce a confound because the number of spikes is also a function of the number of events (see Experimental Procedures).

Figure 3. Reset Task: a variable distractor impairs discrimination of a short but not a long interval

(A) Reset Task. Top rows represent the standard 2T interval discrimination task in a single stimulus protocol. Subjects are asked to press different mouse buttons if they judged the interval to be short (S) or long (L). The feedback across trials results in the creation of an internal representation of the target interval. Bottom rows represent the 3T task in which a distractor is presented at a fixed or variable (dashed) interval across trials. (B) Thresholds for the 100 ms (SHORT) reset task. Left, thresholds for the 100 ms 2T interval discrimination (open bars) and for the 100 ms interval preceded by a distractor presented at the same interval across trials (3T-FIX, red). Right, threshold for the standard 100 ms task (open) and three-tone task in which the distractor was presented at variable intervals across trials (3T-VAR; blue). Error bars represent the SEM. The asterisk represents a significant difference from the other three groups. (C) Reset task (represented as in A), using a 1000 ms (LONG) target interval. Neither of the main effects or the interaction was significant. (D–E) PSE values for the same experiments shown in panels B and C, respectively. The PSE was not significantly different from the target intervals of 100 (D) and 1000 ms (E) in any condition.

Fig 4. Control interval and frequency discrimination tasks

(A) Short-Long reset task. The variable distractor in these trials was between 50–150 ms, and the target interval was 1 s. When a short unpredictable distractor preceded a long target interval there was no effect of whether the distractor was Fixed or Variable. (B) Frequency task. A tone was presented in the absence of a distractor (open bars), or in the presence of a distractor tone presented at a fixed (red) or variable (blue) interval before the target tone. Conventions and color coding as in Figure 3.

Figure 5. Short interstimulus intervals impair interval, but not frequency, discrimination

(A) Bars on the left show the thresholds for a two-interval forced-choice discrimination with a 100 ms target. When the interval between the stimuli was short (250 ms) performance was significantly worse compared to the long interstimulus interval (750 ms). In contrast, performance on a frequency discrimination task was unaltered by the interstimulus interval. (B) Bars on the left illustrate the results for short (250 ms) and long (750 ms) ISI when both the standard and comparison intervals were presented at the same frequency. Bars on the right represent the interval discrimination thresholds when the standard and comparison stimuli were presented at different frequencies. We believe the difference in absolute interval discrimination between both studies (right bars in panels A and B) reflects interference between the different task and stimulus sets in both studies, as well as the inherent subject variability observed in timing tasks.

Figure 6. Dependence of the State-dependent Network on Initial State

(A) Trajectory of the same network shown in Fig. 1 and 2, in response to two 100 ms intervals separated by a 250 (A1) or 750 ms ISI (A2). Note, that the trajectories under the 750 ms ISI are much closer to overlapping than in the 250 ms condition. Arrows indicated the times of the onset of the second interval. (B) Distance matrix. The diagonal represents the distance in Euclidean space between the trajectories shown in A1 and A2 starting at 0. The distance is zero until the onset of the second tone (the noise ‘seed’ was the same for both simulations). The secondary diagonals permit the visualization of the distances between two trajectories shifted in time. This allows the comparison of the trajectory starting at the onset of the second interval (for the 250 ms ISI) and of the first interval (blue rectangle and blue line in lower panel), or the second interval of the 750 ms ISI and the first interval (red rectangle and red line in lower panel). These distances, shown in the lower panel, allows for quantification of the effect of the network not returning to its initial (resting) state before presenting the next stimulus. Note that while the initial distance is lower in the 750 ms ISI, it is not zero. (C) Percent correct performance of networks trained to discriminate two intervals separated by varying ISIs. Average data from four stimulations. Output units were trained to discriminate intervals ranging from 50 – 150 ms. Performance was then tested by examining generalization to these same intervals when presented at varying ISIs after the presentation of a 100 ms interval. Results for the 100 x 150 ms discrimination are shown. Performance is highly dependent on the initial state of the network.