Characterization of Reliability of Spike Timing in Spinal Interneurons During Oscillating Inputs

Abstract

The spike timing in rhythmically active interneurons in the mammalian spinal locomotor network varies from cycle to cycle. We tested the contribution from passive membrane properties to this variable firing pattern, by measuring the reliability of spike timing, P, in interneurons in the isolated neonatal rat spinal cord, using intracellular injection of sinusoidal command currents of different frequencies (0.325–31.25 Hz). P is a measure of the precision of spike timing. In general, P was low at low frequencies and amplitudes ( P = 0–0.6; 0–1.875 Hz; 0–30 pA), and high at high frequencies and amplitudes ( P = 0.8–1; 3.125–31.25 Hz; 30–200 pA). The exact relationship between P and amplitude was difficult to describe because of the well-known low-pass properties of the membrane, which resulted in amplitude attenuation of high-frequency compared with low-frequency command currents. To formalize the analysis we used a leaky integrate and fire (LIF) model with a noise term added. The LIF model was able to reproduce the experimentally observed properties of P as well as the low-pass properties of the membrane. The LIF model enabled us to use the mathematical theory of nonlinear oscillators to analyze the relationship between amplitude, frequency, and P. This was done by systematically calculating the rotational number, N, defined as the number of spikes divided by the number of periods of the command current, for a large number of frequencies and amplitudes. These calculations led to a phase portrait based on the amplitude of the command current versus the frequency-containing areas [Arnold tongues (ATs)] with the same rotational number. The largest ATs in the phase portrait were those where N was a whole integer, and the largest areas in the ATs were seen for middle to high (>3 Hz) frequencies and middle to high amplitudes (50–120 pA). This corresponded to the amplitude- and frequency-evoked increase in P. The model predicted that P would be high when a cell responded with an integer and constant N. This prediction was confirmed by comparing N and P in real experiments. Fitting the result of the LIF model to the experimental data enabled us to estimate the standard deviation of the internal neuronal noise and to use these data to simulate the relationship between N and P in the model. This simulation demonstrated a good correspondence between the theoretical and experimental values. Our data demonstrate that interneurons can respond with a high reliability of spike timing, but only by combining fast and slow oscillations is it possible to obtain a high reliability of firing during rhythmic locomotor movements. Theoretical analysis of the rotation number provided new insights into the mechanism for obtaining reliable spike timing.