An uncertainty principle for neural coding: Conjugate representations of position and velocity are mapped onto firing rates and co-firing rates of neural spike trains.

The hippocampal system contains neural populations that encode an animal's position and velocity as it navigates through space. Here, we show that such populations can embed two codes within their spike trains: a firing rate code ( R) conveyed by within-cell spike intervals, and a co-firing rate code ( ) conveyed by between-cell spike intervals. These two codes behave as conjugates of one another, obeying an analog of the uncertainty principle from physics: information conveyed in R comes at the expense of information in , and vice versa. An exception to this trade-off occurs when spike trains encode a pair of conjugate variables, such as position and velocity, which do not compete for capacity across R and . To illustrate this, we describe two biologically inspired methods for decoding R and , referred to as sigma and sigma-chi decoding, respectively. Simulations of head direction and grid cells show that if firing rates are tuned for position (but not velocity), then position is recovered by sigma decoding, whereas velocity is recovered by sigma-chi decoding. Conversely, simulations of oscillatory interference among theta-modulated "speed cells" show that if co-firing rates are tuned for position (but not velocity), then position is recovered by sigma-chi decoding, whereas velocity is recovered by sigma decoding. Between these two extremes, information about both variables can be distributed across both channels, and partially recovered by both decoders. These results suggest that populations with different spatial and temporal tuning properties-such as speed versus grid cells-might not encode different information, but rather, distribute similar information about position and velocity in different ways across R and . Such conjugate coding of position and velocity may influence how hippocampal populations are interconnected to form functional circuits, and how biological neurons integrate their inputs to decode information from firing rates and spike correlations.