A Unified Mathematical Framework for Coding Time, Space, and Sequences in the Hippocampal Region

Abstract

The medial temporal lobe (MTL) is believed to support episodic memory, vivid recollection of a specific event situated in a particular place at a particular time. There is ample neurophysiological evidence that the MTL computes location in allocentric space and more recent evidence that the MTL also codes for time. Space and time represent a similar computational challenge; both are variables that cannot be simply calculated from the immediately available sensory information. We introduce a simple mathematical framework that computes functions of both spatial location and time as special cases of a more general computation. In this framework, experience unfolding in time is encoded via a set of leaky integrators. These leaky integrators encode the Laplace transform of their input. The information contained in the transform can be recovered using an approximation to the inverse Laplace transform. In the temporal domain, the resulting representation reconstructs the temporal history. By integrating movements, the equations give rise to a representation of the path taken to arrive at the present location. By modulating the transform with information about allocentric velocity, the equations code for position of a landmark. Simulated cells show a close correspondence to neurons observed in various regions for all three cases. In the temporal domain, novel secondary analyses of hippocampal time cells verified several qualitative predictions of the model. An integrated representation of spatiotemporal context can be computed by taking conjunctions of these elemental inputs, leading to a correspondence with conjunctive neural representations observed in dorsal CA1.

Entorhinal–hippocampal neuronal circuits bridge temporally discontiguous events

“…5). This idea is also supported by a computational study on the contribution of persistent firing on time-cell activity (Hasselmo 2012;Hasselmo and Stern 2014;Howard et al 2014;Saravanan et al 2015). In this way, persistent firing is transmitted to CA1 pyramidal cells to bridge the temporal gap, and then to the amygdala via the EC layer V to coincide with the onset of the US (Fendt and Fanselow 1999) to generate a fear memory engram via Hebbian synaptic strengthening in the amygdala.…”

Neural Mechanism to Simulate a Scale-Invariant Future

“…1b) where the first layer encodes the Laplace transform of externally observed stimuli in real time, and the second layer approximately inverts the Laplace transform to represent a fuzzy estimate of the actual stimulus history [14]. With access to instantaneous velocity of motion, this two layer network representing temporal memory can be straightforwardly generalized to represent onedimensional spatial memory [15]. Hence in the context of this two layer network, time-translation of the temporal memory representation can be considered mathematically equivalent to space-translation of the spatial memory representation.…”

“…In this section we start with a basic overview of the two layer memory model and summarize the relevant details from previous work [14,15,22] to serve as a background. Following that, we derive the equations that allow the memory nodes to be coherently time-translated to various future moments in synchrony with the theta oscillations.…”

“…We emphasize that although the development in the preceding section was done with respect to time, all of the results generalize to one-dimensional position as well (eq. 5, [15]). The overwhelming majority of evidence for phase precession comes from studies of place cells (but see [3]).…”

If I had a million neurons

“…In this framework, each event in the environment activates a trace that decays exponentially. The inverse transform of these traces across a population of neurons can drive output of spiking at a specific temporal interval (Howard et al, 2014a) with a time course that widens with temporal delay in a manner similar to the experimental data.…”