Fast estimation of phase-resetting curves in real neurons

Consequences for the computational modeling of biological neural networks

Neurons that fire regularly can be mathematically described as oscillators. Under fairly general conditions the dynamics of an oscillator, and in particular of a neuron, are determined by the phase-resetting curve (PRC; or phase-response curve). The PRC is a model-independent description of the effect of perturbations onto the periodic motion of an oscillator: It quantifies how much the period of the oscillator increases or decreases in response to a perturbation occurring at any phase.

For neural oscillators (therefore excluding "bursting neurons", which are not fully described by a phase variable) the PRC reveals two possible qualitative behaviors in response to synaptic inputs: i) integrators, or type I neurons and ii) resonators, or type II neurons. Interestingly, type I neurons possess nonnegative PRC whereas type II neural oscillators possess PRC that are partially positive and partially negative. That is, an infinitesimal positive perturbation of the membrane potential will never delay the next spike in a type I neuron, whereas such a perturbation will delay or advance the next spike in a type II neuron, depending on the phase at which the pulse is delivered.

An empirical, reliable estimation of neuronal PRC presents difficulties that have impaired neuroscientists from taking systematically advantage of the information PRC provide: for example, the PRC can be used to predict features of the network dynamics like the formation of synchronized neural assemblies. Combining theory, computer simulations and electrophysiological recordings (whole-cell patch-clamp) we have developed a successful approach to estimating the PRC of real neurons easily. This allows us to simplify the complex dynamics of a single neuron into a reduced one-dimensional phase model without losing relevant information. Details of our method are provided in R.F.Galán et al. (2005) Phys. Rev. Lett. 94, 158101 [PDF]. A Matlab implementation of our PRC estimator can be downloaded here [zip file].

Figure: Estimated phase-resetting curve of a mitral cell. The curve clearly shows that the neuron behaves as a resonator (type II neuron). Further mathematical analysis using this PRC reveals how mitral cells are capable of synchronizing through inhibitory, but not excitatory synapses.