A review of a preprint by Y. Ahmadian and K. D. Miller

What is the dynamical regime of cortical networks? This question has been debated as long as we have been able to measure cortical activity. The question itself can be interpreted in multiple ways, the answers depending on spatial and temporal scales of the activity, behavioral states, and other factors. Moreover, we can characterize dynamics in terms of dimensionality, correlations, oscillatory structure, or other features of neural activity.

In this review/comment, Y. Ahmadian and K. Miller consider the dynamics of single cells in cortical circuits, as characterized by multi-electrode, and intracellular recording techniques. Numerous experiments of this type indicate that individual cells receive excitation and inhibition that are approximately balanced. As a result, activity is driven by fluctuations that cause the cell’s membrane potential to occasionally, and irregularly cross a firing threshold. Moreover, this balance is not a result of a fine tuning between excitatory and inhibitory weights, but is achieved dynamically.

There have been several theoretical approaches to explain the emergence of such balance. Perhaps the most influential of these theories was developed by C. van Vreeswijk and H. Sompolinsky in 1996. This theory of dynamic balance relies on the assumption that the number of  excitatory and inhibitory inputs to a cell, K, is large and that these inputs scale like 1/\sqrt(K). If external inputs to the network are strong, under fairly general conditions activity is irregular, and in a balanced regime: The average difference between the excitatory and inhibitory input to a cell is much smaller than either the excitatory input or inhibitory input itself. Ahmadian and Miller refer to this as the tightly balanced regime.

In contrast, excitation and inhibition still cancel approximately in loosely balanced networks. However, in such networks the residual input is comparable to the excitatory input, and cancellation is thus not as tight. This definition is too broad, however, and the authors also assume that the net input (excitation minus inhibition) driving each neuron grows sublinearly as a function of the external input. As shown previously by the authors and others, such a state emerges when the number of inputs to each cell is not too large, and each cell’s firing rate grows superlinearly with input strength. Under these conditions a sufficiently strong input to the network evokes fast inhibition that loosely balances excitation to prevent runaway activity.

Loose and tight balance can occur in the same model network, but loose balance occurs at intermediate external inputs, while tight balance emerges at high external input levels. While the transition between the two regimes is not sharp, the network behaves very differently in the two regimes: A tightly balanced network responds linearly to its inputs, while the response of a loosely balanced network can be nonlinear. Moreover, external input can be of the same order as the total input for loosely balanced networks, but must be much larger than the total input (of the same order as excitation and inhibition on their own) for tightly balanced networks.

Which of these two regimes describe the state of the cortex? Tightness of balance is difficult to measure directly, as one cannot isolate excitatory and inhibitory inputs to the same cell simultaneously. However, the authors present a number of strong, indirect arguments in favor of loose balance basing their argument on several experimental findings: 1) Recordings suggest that the ratio of the mean to the standard deviation excitatory input is not sufficiently large to necessitate precise cancellation from inhibition. This would put the network in the loosely balanced regime. Moreover, excitatory currents alone are not too strong, comparable to the voltage difference between the mean and threshold. 2) Cooling and silencing studies suggest that external input, e.g. from thalamus, to local cortical networks is comparable to the net input. This is consistent with loose balance, as tight balance is characterized by strong external inputs. 3) Perhaps most importantly cortical computations are nonlinear. Sublinear response summation, and surround suppression, for instance, can be implemented by loosely balanced networks. However, classical tightly balanced networks exhibit linear responses, and thus cannot implement these computations. 4) Tightly balanced networks are uncorrelated, and do not exhibit the stimulus modulated correlations observed in cortical networks.

These observations deserve a few comments: 1) The transition from tight to loose balance is gradual. It is therefore not exactly clear when, for instance, the mean excitatory input is sufficiently strong to require tight cancellation. As the authors suggest, some cortical areas may therefore lean more towards a tight balance, while others lean more towards loose balance. 2) It is unclear whether cooling reduces inputs to the cortical areas in question. 3 and 4) Classical tightly balanced networks are unstructured and are driven by uncorrelated inputs. Changes to these assumptions can result in networks that do exhibit a richer dynamical repertoire including, spatiotemporally structured, and correlated behaviors, as well as nonlinear computations.

Why does this this debate matter? The dynamical regime of the cortex describes how a population of neurons transforms its inputs, and thus the computations that a network can perform. The questions of which computations the cortex performs, and how it does so, are therefore closely related to questions about its dynamics. However, at present our answers are somewhat limited. Most current theories ignore several features of cortical networks that may impact their dynamics: There is a great diversity of cell types, particularly inhibitory cells, each with its own dynamical, and connectivity properties. It is likely that this diversity of cells shape the dynamical state of the network in a way that we do not yet fully understand. Moreover, the distribution of synaptic weights, and spatial synaptic interactions across the dendritic trees are not accurately captured in most models. It is possible, that these, and other, details are irrelevant, and current theories of balance are robust. However, this is not yet completely clear. Thus, while the authors make a strong case that the cortex is loosely balanced, a definitive answer to this question lies in the future.

Thanks go to Robert Rosenbaum for his input on this post.