Combining synaptic and cellular resonance in a feed-forward neuronal network

Abstract

We derive a mathematical theory to explain the subthreshold resonance response of a neuron to synaptic input. The theory shows how a neuron combines information from its intrinsic resonant properties with those of the synapse to determine the neuron's generalized resonance response. Our results show that the maximal response of a postsynaptic neuron can lie between the preferred intrinsic frequency of the neuron and the synaptic resonance frequency. We compare our theoretical results to parallel findings on experiments of the crab pyloric central pattern generator.

Figure 1

Measurement of CRC, SRC and GRC in the crab pyloric network. A. The CRC of the LP neuron was measured by injection a ZAP current of amplitude +/−2 nA sweeping frequencies between 0.1 and 5 Hz and indicated a resonance frequency at 1.9 Hz. Inset shows schematic diagram indicating the synaptic connection from the PD neuron to the LP neuron. B. The SRC was measured by voltage-clamping the PD neuron with a ZAP function waveform and measuring the synaptic current in the LP neuron. The LP neuron was voltage clamped at a holding potential of −60 mV. The SRC showed a peak value at 0.3 Hz. C. The GRC was measured by injecting a ZAP function waveform into the voltage-clamped PD neuron while the postsynaptic response of the LP neuron was measured in current clamp. The GRC showed a peak value at 0.6 Hz D. The CRC, SRC and GRC plotted as a function of frequency indicate the distinct peaks of the CRC and the SRC and that the GRC peak falls between these two values.

Figure 2

The response curves corresponding to equations (1–2). The CRCs (dotted curves) are the amplitude of oscillations for the given frequency (horizontal axis). The GRCs (solid curves) are the oscillation amplitude of the postsynaptic response when the signal is generated using a resonant synapse. The SRCs (dashed) are unitless curves superimposed so that the preferred resonant frequency can be seen. The maximum network response occurs for a frequency that is neither the resonant frequency for the cell nor the preferred frequency for the synapse. In the left panel σ = 10 ms², resulting in a steeper profile for the SRC than in the right panel. The peak of the GRC is located closer to the peak of the SRC than in the right panel (σ = 40 ms²).

Figure 3

The SRC, CRC, and GRC when the synapse described by (3) (4) is used. The Hodgkin-Huxley parameters are the same as in Fig. 2. Parameter values for the synapse in this figure are τ_rec = 125, τ₁ = 3, τ_facil = 100 and U = 0.01. The peak of the SRC is hard to discern in the figure, but lies near 130 Hz.