A1 Functional advantages of cell-type heterogeneity in neural circuits

In a recent paper the authors developed a stochastic model for the response of the cerebral cortex to a general anesthetic agent. The model predicted that there would be an anesthetic-induced phase change at the point of transition into unconsciousness, manifested as a divergence in the electroencephalogram spectral power, and a change in spectral energy distribution from being relatively broadband in the conscious state to being strongly biased towards much lower frequencies in the unconscious state. Both predictions have been verified in recent clinical measurements. In the present paper we extend the model by calculating the equilibrium distribution function for the cortex, allowing us to establish a correspondence between the cortical phase transition and the more familiar thermodynamic phase transitions. This correspondence is achieved by first identifying a cortical free energy function, then by postulating that there exists an inverse relationship between an anesthetic effect and a quantity we define as cortical excitability, which plays a role analogous to temperature in thermodynamic phase transitions. We follow standard thermodynamic theory to compute a cortical entropy and a cortical “heat capacity,” and we investigate how these will vary with anesthetic concentration. The significant result is the prediction that the entropy will decrease discontinuously at the moment of induction into unconsciousness, concomitant with a release of “latent heat” which should manifest as a divergence in the analogous heat capacity. There is clear clinical evidence of heat capacity divergence in historical anesthetic-effect measurements performed in 1977 by Stullken et al. [Anesthesiology 46, 28 (1977)]. The discontinuous step change in cortical entropy suggests that the cortical phase transition is analogous to a first-order thermodynamic transition in which the comatose-quiescent state is strongly ordered, while the active cortical state is relatively disordered.

One of the grand puzzles in neuroscience is establishing the link between cognition and the disparate patterns of spontaneous and task-induced brain activity that can be measured clinically using a wide range of detection modalities such as scalp electrodes and imaging tomography. High-level brain function is not a single-neuron property, yet emerges as a cooperative phenomenon of multiply-interacting populations of neurons. Therefore a fruitful modeling approach is to picture the cerebral cortex as a continuum characterized by parameters that have been averaged over a small volume of cortical tissue. Such mean-field cortical models have been used to investigate gross patterns of brain behavior such as anesthesia, the cycles of natural sleep, memory and erasure in slow-wave sleep, and epilepsy. There is persuasive and accumulating evidence that direct gap-junction connections between inhibitory neurons promote synchronous oscillatory behavior both locally and across distances of some centimeters, but, to date, continuum models have ignored gap-junction connectivity. In this paper we employ simple mean-field arguments to derive an expression for $D_2$, the diffusive coupling strength arising from gap-junction connections between inhibitory neurons. Using recent neurophysiological measurements reported by Fukuda et al. [J. Neurosci. 26, 3434 (2006)], we estimate an upper limit of $D_2\approx 0.6{\mathrm{cm}}^2$. We apply a linear stability analysis to a standard mean-field cortical model, augmented with gap-junction diffusion, and find this value for the diffusive coupling strength to be close to the critical value required to destabilize the homogeneous steady state. Computer simulations demonstrate that larger values of $D_2$ cause the noise-driven model cortex to spontaneously crystalize into random mazelike Turing structures: centimeter-scale spatial patterns in which regions of high-firing activity are intermixed with regions of low-firing activity. These structures are consistent with the spatial variations in brain activity patterns detected with the BOLD (blood oxygen–level–dependent) signal detected with magnetic resonance imaging, and may provide a natural substrate for synchronous gamma-band rhythms observed across separated EEG (electroencephalogram) electrodes.

Understanding the structure and purpose of sleep remains one of the grand challenges of neurobiology. Here we use a mean-field linearized theory of the sleeping cortex to derive statistics for synaptic learning and memory erasure. The growth in correlated low-frequency high-amplitude voltage fluctuations during slow-wave sleep (SWS) is characterized by a probability density function that becomes broader and shallower as the transition into rapid-eye-movement (REM) sleep is approached. At transition, the Shannon information entropy of the fluctuations is maximized. If we assume Hebbian-learning rules apply to the cortex, then its correlated response to white-noise stimulation during SWS provides a natural mechanism for a synaptic weight change that will tend to shut down reverberant neural activity. In contrast, during REM sleep the weights will evolve in a direction that encourages excitatory activity. These entropy and weight-change predictions lead us to identify the final portion of deep SWS that occurs immediately prior to transition into REM sleep as a time of enhanced erasure of labile memory. We draw a link between the sleeping cortex and Landauer’s dissipation theorem for irreversible computing [R. Landauer, IBM J. Res. Devel. 5, 183 (1961)], arguing that because information erasure is an irreversible computation, there is an inherent entropy cost as the cortex transits from SWS into REM sleep.