Mathematics informs optogenetic stimulation of the brain

Optogenetics has given neurophysiologists the remarkable ability to directly stimulate neurons with light. The method works by transducing neurons with a virus that contains photo-sensitive proteins (opsins). It has become a popular method for probing neural function in living animals. But does optogenetically stimulated tissue operate with normal neuronal dynamics? Recent observations in macaque motor cortex (Lu et al, 2015) provide us with a surprising clue that it is only apparent with mathematical neuroscience.

Lu et al (2015) transduced a small patch of cortex and recorded the local field potential in the immediate vicinity. They found that constant optical stimulation elicited 40-80 Hz oscillations in the local field potential that propagated far into the surrounding cortical tissue, well beyond the reach of the optical stimulation. They correctly recognised that those oscillations emerged in a manner consistent with a dynamical system undergoing a supercritical Hopf bifurcation. However they did not recognise that such dynamics rarely support propagating waves.

Adapted from Lu et al (2015) A. Spectral power of gamma-band oscillations in the local field potential of optogenetically-transduced neural tissue versus the intensity of optical stimulation. B. Expanding rings of gamma-band oscillations in the local field potential recorded by a 4x4 mm microelectrode array.

The supercritical Hopf bifurcation is the hallmark of what is known in mathematical neuroscience as Type II neural excitability. Type II neurons rarely sustain propagating waves because they fire low-amplitude spikes in response to weak stimulation.  Type I neurons, on the other hand, do sustain propagating waves because they always fire large-amplitude spikes even if the stimulation is very weak. So how can cortical tissue simultaneously exhibit both Type I and Type II excitability as the observations by Lu et al (2015) suggest?

We investigated the apparent contradiction by modelling the cortex as recurrently-connected populations of excitatory and inhibitory neurons using the classic Wilson-Cowan model. Such models exhibit either Type I or Type II excitability depending upon the choice of parameters. Our insight was that optogenetic stimulation can locally transform Type I excitability into Type II excitability by early activation of the inhibitory neurons. In this way, the non-stimulated regions of cortex retain support for propagating waves while the stimulated tissue exhibits supercritical Hopf dynamics.

Space-time plots of the cortical model in one spatial dimension. Optogenetic stimulation was applied focally at position x=0.  Panels A-C show cases of weak, medium, and strong stimulation respectively. Gamma oscillations arose gradually from the stimulation site via a supercritical Hopf bifurcation. Waves were emitted from the stimulation site once the simulation reached a critical threshold. The medium thus exhibited co-existing Type I and Type II excitability.
Our model shows how optogenetic stimulation can alter the excitability of neural tissue. The findings have implications for neurophysiological experiments which assume that optogenetically stimulated tissue operates with the same dynamics as endogenous neural activity.


Heitmann, Rule, Truccolo, Ermentrout (2017) Optogenetic stimulation shifts the excitability of cerebral cortex from Type I to Type II: Oscillation onset and wave propagation. PLOS Computational Biology. 13(1): e1005349. doi: 10.1371/journals.pcbi.1005349

Lu, Truccolo, Wagnerm Varas-Irwin, Ozden, Zimmermann, May, Agha, Wang, Nurmikko (2015) Optogenetically induced spatiotemporal gamma oscillations and neuronal spiking activity in primate motor cortex. J Neurophysiol 113: 3574-3587.