Neuronal Oscillators in Aplysia californica that Demonstrate Weak Coupling In Vitro

Networks of oscillators produce vital activity in diverse natural systems. The dynamics of these networks are frequently studied via computational models that assume weak coupling, yet this assumption has not been experimentally validated. We applied weak stimuli to neuronal oscillators in Aplysia californica and deconvolved infinitesimal phase response curves (IPRCs) that describe the phase response of a neuron. We show that these IPRCs reliably predict the phase response for weak stimuli, independent of the stimulus waveform used. These weak stimuli are in the range of normal synaptic activity for these neurons, suggesting that weak coupling is a likely mechanism.

Figure 1

(a) Block diagram of the experimental setup. (b) Sample voltage traces showing a typical period of the neuronal oscillator and a perturbed period with the stimulus applied at $t_s$.

Figure 2

Weak amplitude stimuli elicit PRCs that scale linearly with amplitude, while large stimuli do not. PRCs were obtained from the same neuron for a wide range of stimulus amplitudes, for $\tau =10\mathrm{ms}$. Cobalt chloride saline was used to eliminate synaptic input. (a) Weak stimuli PRCs measured for conductances ranging from $0.02\mu \mathrm{S}–0.1\mu \mathrm{S}$. (b) Strong stimuli PRCs for conductances from $0.1\mu \mathrm{S}–0.6\mu \mathrm{S}$. (c) Weak stimuli PRCs scaled, point by point, to a conductance of $0.1\mu \mathrm{S}$. (d) Strong stimuli PRCs scaled to a conductance of $0.6\mu \mathrm{S}$.

Figure 3

The IPRCs and reconstructed PRCs for 3 different experiments. (a)–(c) Each show three IPRCs recovered for stimuli with $\tau =10$, 20, and 40 ms. (d)–(f) The response functions convolved with stimulus waveforms are better able to reproduce PRC data for weak amplitude PRCs. Data points are the measured PRCs in response to stimuli with $\tau =20\mathrm{ms}$. The orange line is a spline fit to the PRC data points, red, green, and blue lines are the convolution of the IPRCs shown in panels (a)–(c) with the $\tau =20\mathrm{ms}$ stimulus waveform. The rows correspond to different experiments with a set conductance ($g$) and maximum PRC amplitude of the 10 ms PRC ($\Delta \theta$) of (1) $g=0.005\mu \mathrm{S}$, $\Delta \theta =0.038$; (2) $g=0.04\mu \mathrm{S}$, $\Delta \theta =0.095$; (3) $g=0.12\mu \mathrm{S}$, $\Delta \theta =0.694$.

Figure 4

There is a several order-of-magnitude reduction in MSE for low amplitude PRCs. MSE is plotted as a function of PRC amplitude across all experiments. Red, green, and blue points represent MSEs of reconvolved PRCs for IPRCs in response to $\tau =10$, 20, or 40 ms stimuli, respectively. IPRCs are convolved with stimulus waveforms and compared to the spline fit PRC data. Panels correspond to stimulus waveforms. (a) $\tau =10\mathrm{ms}$ stimulus. (b) $\tau =20\mathrm{ms}$ stimulus. (c) $\tau =40\mathrm{ms}$ stimulus.