JTW's Evolutionary Origins - Author: Edelman, Gerald M.

A Neural Complexity Measure

Theoretical Concepts and Measures

This brief review of neurological and neurophysiological data indicates that the distributed neural process underlying conscious experience must be functionally integrated and at the same time highly differentiated.

As mentioned above, two key properties of conscious experience are that it is integrated, in the sense that it cannot be subdivided into independent components, and that it is extremely differentiated, in the sense that it is possible, within a short time, to select among an enormous number of different conscious states.

It is a central claim of this article that analyzing the convergence between these phenomenological and neural properties can yield valuable insights into the kinds of neural processes that can account for the corresponding properties of conscious experience.

Such an analysis requires the availability of satisfactory measures of integration and differentiation that can be applied to actual neural processes, as well as an understanding of the neural mechanisms of integration.

Functional clustering:
How to identify an integrated process.

How can one determine whether a neural process is unified or simply a collection of independent or nearly independent subprocesses?

We have suggested that a subset of distributed elements within a system gives rise to a single, integrated process if, at a given time scale, these elements interact much more strongly among themselves than with the rest of the system --for example, if they form a functional cluster.

This criterion has been formalized by introducing a direct measure of functional clustering (35) which we summarize here.

Consider a j^th subset of k elements (X^k_j) taken from an isolated neural system X, and its complement (X - X^k_j).

Interactions between the subset [X^k_j] and the rest of the system [(X - X^k_j)] introduce statistical dependence between the two.

This is measured most generally by their mutual information

MI(X^k_j; X - X^k_j) = H(X^k_j) + H(X - X^k_j) - H(X),

which captures the extent to which the entropy of X^k_j is accounted for by the entropy of X - X^k_j and vice versa [H indicates statistical entropy (36)].

The statistical dependence within a subset can be measured by a generalization of mutual information, which is called integration [I] and is given by

I(X^k_j) = ΣH(x_i) - H(X^k_j),

where H(x_i) is the entropy of each element x_i considered independently.

We then define the functional cluster index [CI]

CI(X^k_j) = I(X^k_j)/MI(X^k_j; X - X^k_j)

as a ratio of the statistical dependence within the subset [I] and the statistical dependence between that subset and the rest of the system [MI].

Based on this definition, a subset of neural elements that has a CI value much higher than 1 and does not itself contain any smaller subset with a higher CI value constitutes a functional cluster.

[if CI > 1 then subset X^k_j is a Functional Cluster].

This is a single, integrated neural process that cannot be decomposed into independent or nearly independent components.

We have applied these measures of functional clustering both to simulated datasets and to positron emission tomography data obtained from schizophrenic subjects performing cognitive tasks (35).

Theoretically sound measures that can detect the occurrence of functional clustering at the time scale (fractions of a second) crucial for conscious experience may require additional assumptions.

Nevertheless, it would appear that the rapid establishment of synchronous firing among cortical regions and between cortex and thalamus should be considered as an indirect indicator of functional clustering, since it implies strong and fast neural interactions among the neural populations involved (19,20).

The mechanisms of rapid functional clustering among distributed populations of neurons in the thalamocortical system have been studied with the help of large-scale simulations (19,20).

These have shown that the emergence of high-frequency synchronous firing in the thalamocortical system depends critically on the dynamics of corticothalamic and corticocortical reentrant circuits and on the opening of voltage-dependent channels in the horizontal corticocortical connections (37).

Neural complexity:
Measuring the differences that make a difference.

Once an integrated neural process is identified, we need to determine to what degree that process is differentiated.

Does it give rise to a large repertoire of different activity patterns or neural states?

It is essential to consider only those differences between activity patterns that make a difference to the system itself.

A TV screen may, for example, go through a large number of "activity patterns" that look different to an external observer, but that make no difference to the TV.

A possible approach to measuring differences that make a difference within an integrated neural system is to consider it as its own "observer."

This can be achieved by dividing the system (which, we assume, constitutes a functional cluster) into two subsets and then measuring their mutual information (38).

The value of MI(X^k_j; X - X^k_j) between a j^th subset (X^k_j) of the isolated system X and its complement X - X^k_j will be high if two conditions are met.

Both X^k_j and X - X^k_j must have many states [their entropy must be relatively high (10)], and the states of X^k_j and of X - X^k_j must be statistically dependent (the entropy of X^k_j must be largely accounted for by the interactions with X - X^k_j, and vice versa).

The expression MI(X^k_j; X - X^k_j) reflects how much, on average, changes in the state of X - X^k_j make a difference to the state of X^k_j , and vice versa.

To obtain an overall measure of how differentiated a system is, one can consider not just a single subset of its constituent elements, but all its possible subsets.

The corresponding measure, called neural complexity [CN], is given by

CN(X) = 1/2 ΣMI(X^k_j; X - X^k_j),

where the sum is taken over all k subset sizes and the average is taken over all j^th combinations of k elements.

Complexity is thus a function of the average mutual information between each subset and the rest of the system, and it reflects the number of states of a system that result from interactions among its elements (39).

It can be shown that high values of complexity reflect the coexistence of a high degree of functional specialization and functional integration within a system, as appears to be the case for systems such as the brain.

For example, the dynamic behavior of a simulated cortical area containing thousands of spontaneously active neuronal groups (38) resembled the low-voltage fast-activity EEG of waking states and had high complexity.

Such a system, whose connections were organized according to the rules found in the cortex, visited a large repertoire of different activity patterns that were the result of interactions among its elements.

If the density of the connections was reduced, the dynamic behavior of the model resembled that of a noisy TV screen and had minimal complexity.

A large number of activity patterns were visited, but they were merely the result of the independent fluctuations of its elements.

If the connections within the cortical area were instead distributed at random, the system yielded a hypersynchronous EEG that resembled the high-voltage waves of slow-wave sleep or of generalized epilepsy.

The system visited a very limited repertoire of activity patterns, and its complexity was low.

Measures of complexity, like measures of functional clustering, can also be applied to neurophysiological data to evaluate the degree to which a neural process is both integrated and differentiated (40).

This opens the way to comparisons of the values of neural complexity in different cognitive and arousal states and to empirical tests of the relationships between brain complexity and conscious experience.