Navigating Through Time: A Spatial Navigation Perspective on How the Brain May Encode Time

Abstract

Interval timing, which operates on timescales of seconds to minutes, is distributed across multiple brain regions and may use distinct circuit mechanisms as compared to millisecond timing and circadian rhythms. However, its study has proven difficult, as timing on this scale is deeply entangled with other behaviors. Several circuit and cellular mechanisms could generate sequential or ramping activity patterns that carry timing information. Here we propose that a productive approach is to draw parallels between interval timing and spatial navigation, where direct analogies can be made between the variables of interest and the mathematical operations necessitated. Along with designing experiments that isolate or disambiguate timing behavior from other variables, new techniques will facilitate studies that directly address the neural mechanisms that are responsible for interval timing.

Keywords: animal behavior; episodic memory; interval timing; neural circuits; spatial navigation; timing.

Figure 1:. Scales of neural timing and analogy to spatial navigation.

(a) An animal is foraging for food when a predator is detected. The animal can choose to wait for a given amount of time (top) or risk continuing without waiting (bottom). (b) Diversity of timescales, brain regions, and neural circuits involved in the encoding of time from milliseconds to days. (far left) Neurons in auditory cortex tuned to msec inter-click intervals (Sadagopan & Wang 2009). (middle left) In eyeblink conditioning, a conditioned stimulus (tone) precedes an unconditioned stimulus (air puff) by a fixed interval, and animals learn to blink and cerebellar neurons respond just before the air puff (Berthier & Moore 1986; Kotani et al. 2003). (near left) Two odors are presented with a delay, and the animal licks to receive a reward only when the odors are identical. Pyramidal cells of hippocampus fire sequentially during the delay period (MacDonald et al. 2013). (near right) Animals press a lever after a delay period to receive a reward. Neurons in cortex and striatum encode the delay period with a variety of patterns including tuning to various times or ramping activity (Matell et al. 2003). (middle right) Neurons in the LEC encode the cumulative duration of traversals in alternating black and white rooms over the course of tens of minutes (Tsao et al. 2018). (far right) A transcription-translation feedback loop in neurons of the suprachiasmatic nucleus results in 24-hour rhythms (Takahashi 2017; Welsh et al. 1995). Illustrations inspired by data from the cited studies. (c) Spatial path integration in 2-D (left half). At ‘start,’ position is known exactly. As the animal moves towards target ‘end’ or goal, actual position x(t) is shown in black and internal estimate of position x’(t) in gray. Integration of a velocity signal (yellow region, inferred from self-motion cues) updates position but accumulates error until landmark or border that allows for error correction is reached (‘reset’ or ‘end’, gray arrow). Cells in spatial navigation (right half), from left to right: velocity cells and head direction cells encode speed and orientation of the animal with uncertainty indicated in yellow; grid cells encode periodic representation of position; place cells encode current position; border cells indicate proximity to borders or objects. Grid cells integrate incoming speed and orientation signals and are summed to produce single-peaked place cells. Border cells can reset or correct these position signals. (d) Interval timing through time-integration in 1-D (left half). Elapsed time is measured after a start signal. External time t shown in black while internal estimate of time t’, in gray, indicates perception that more time (gray line above black line) or less time (gray line below black line) has elapsed relative to external time. Estimate of time is informed by a moment-by-moment temporal derivative that represents the animal’s ongoing passage of time signal and runs until a ‘reset’ (correction cue) or ‘stop’ is encountered. Cells of interval timing (right half): temporal derivative cell encodes internal representation of the passage of time, represented here as a pacemaker (left) whose inter-spike-interval dt’ approximates discrete time steps dt. Continuous-time version of this process is represented by temporal derivative cell whose deviation from true time (black line) is shown by the gray trace. Output of temporal derivative cell is integrated by hypothesized periodically firing temporal cells. These are summed to produce well-tuned time cells. Border cells fire at start or stop positions and reset or correct encoding of time in temporal periodic cells and in time cells.

Figure 2:. Models for time cells.

(a) Sequential firing of time cells. Each row represents firing rate (FR) of one cell. Each cell is active at a specific moment within the time interval, and together the sequence of firing cells tiles the entire interval. (b) Ring attractor CAN model. Start population of cells fire a burst at the beginning of the time interval (left, green) and input to a ring attractor (middle, black). Global inhibition provides end signal (right, red). (c) Line attractor CAN model. Start population fire a burst at the beginning of the time interval (left, green) and input to a line attractor (middle, black). End signal is simply cells at the end of the sequence (right, red). (d) Mean connectivity pattern of cells in the CAN model. y-axis represents strength of outgoing weights. x-axis represents time preference of cells that receive the inputs (aligned to the time preference of the output cell at 0, dashed line). (e) Sequence of 2 time cells created by the oscillatory interference model. Start cell fires during the entire interval (top left, green). End signal cell inhibits start cell or integrator cells at the end of the interval (left bottom). On the right a scheme of 2 cells (one in black and one in grey). Each cell excited by a start cell. The dendritic frequency oscillation of each cell (F_dendrite) is different than its somatic frequency oscillation (F_soma, which differs in the 2 cells) and results in an interference pattern (Vm). The membrane potential crosses the firing threshold every beat oscillation. (f) Scheme of 3 ramping cells. Each cell has a different slope. (g) A network model that uses a positive feedback mechanism to integrate inputs over time and generate ramping cells. All integrator cells receive feedforward input from start cells (green). Output integrator cells (black) are connected by recurrent excitation and also send excitation to end neurons. End cells (red) inhibit integrator cells once the rate of inputs they receive crosses a threshold or by external input. Adapted from (Seung et al. 2000). (h) Activity of each cell type (start cells: green; end cells: red; integrator cells: black) when recurrent excitation is equal to the leak current (left, W_pos > Leak). or when recurrent excitation is slightly higher than the leak current (right, W_pos > Leak). (i) A model network generating ramping activity by an adaptation process. A population of excitatory persistent firing cells (fire at a constant rate during the time interval) excite a population of inhibitory (Inh) and excitatory (Ex) neurons. Due to adaptation of the firing rate in the inhibitory neurons, their firing rate decreases. This disinhibits the excitatory population, which in turn increases their firing rate, which creates a ramping firing pattern. Adapted from (Reutimann et al. 2004). (j) Exponentially decaying firing rates of 6 cells with different time constants in response to a start burst, analogous to the Laplace transform of a delta function (a start burst). (k) Start cells (green) give a pulse input to a cell population with exponentially decaying firing rates, each with different time constants (the cells arranged according to their time constants in increasing order bottom to top). The exponentially decaying cell population projects to the output layer cells (on the right) through center-off surround-on connectivity pattern. Thus, each cell in the output layer receives a weighted sum of several exponential decaying cells, implementing an inverse Laplace transform, which in turn produces time cells in the output layer (Howard et al. 2014; Liu et al. 2019).