The phase response of the cortical slow oscillation

Abstract

Cortical slow oscillations occur in the mammalian brain during deep sleep and have been shown to contribute to memory consolidation, an effect that can be enhanced by electrical stimulation. As the precise underlying working mechanisms are not known it is desired to develop and analyze computational models of slow oscillations and to study the response to electrical stimuli. In this paper we employ the conductance based model of Compte et al. (J Neurophysiol 89:2707–2725, 2003) to study the effect of electrical stimulation. The population response to electrical stimulation depends on the timing of the stimulus with respect to the state of the slow oscillation. First, we reproduce the experimental results of electrical stimulation in ferret brain slices by Shu et al. (Nature 423:288–293, 2003) from the conductance based model. We then numerically obtain the phase response curve for the conductance based network model to quantify the network’s response to weak stimuli. Our results agree with experiments in vivo and in vitro that show that sensitivity to stimulation is weaker in the up than in the down state. However, we also find that within the up state stimulation leads to a shortening of the up state, or phase advance, whereas during the up–down transition a prolongation of up states is possible, resulting in a phase delay. Finally, we compute the phase response curve for the simple mean-field model by Ngo et al. (EPL Europhys Lett 89:68002, 2010) and find that the qualitative shape of the PRC is preserved, despite its different mechanism for the generation of slow oscillations.

Appendix: The network model

In the original model by (Compte et al. 2003) 1,024 pyramidal neurons (see Table 1) and 256 interneurons (see Table 2) are distributed equidistantly along a line of 5 mm. The probability that two neurons, separated by a distance x, are connected is $P(x)=\left(\frac{1} {\sqrt{2\pi \sigma^2}}\right)\exp(-x^2/2\sigma^2)$ with a synaptic footprint of $\sigma = 250\,\upmu\hbox{m}$ for excitatory connections and $\sigma=125\,\upmu\hbox{m}$ for inhibitory connections. The equations governing the synapses can be found in Table 3. Each neuron makes 20 ± 5 connections to other neurons. In our simulations we used 256 pyramidal neurons and 64 interneurons. The network length and synaptic footprint was linearly scaled to preserve the properties of the original model. We applied periodic boundary conditions.

Table 1 Regular-spiking pyramidal neurons

Table 2 Fast-spiking inhibitory interneurons

Table 3 Synapses