Figure 1.

Effective information, minimum information bipartition, and complexes. a. Effective information. Shown is a single subset S of 4 elements ({1,2,3,4}, blue circle), forming part of a larger system X (black ellipse). This subset is bisected into A and B by a bipartition ({1,3}/{2,4}, indicated by the dotted grey line). Arrows indicate causally effective connections linking A to B and B to A across the bipartition (other connections may link both A and B to the rest of the system X). To measure EI(A→B), maximum entropy Hmax is injected into the outgoing connections from A (corresponding to independent noise sources). The entropy of the states of B that is due to the input from A is then measured. Note that A can affect B directly through connections linking the two subsets, as well as indirectly via X. Applying maximum entropy to B allows one to measure EI(B→A). The effective information for this bipartition is EI(A B) = EI(A→B) + EI(B→A). b. Minimum information bipartition. For subset S = {1,2,3,4}, the horizontal bipartition {1,3}/{2,4} yields a positive value of EI. However, the bipartition {1,2}/{3,4} yields EI = 0 and is a minimum information bipartition (MIB) for this subset. The other bipartitions of subset S = {1,2,3,4} are {1,4}/{2,3}, {1}/{2,3,4}, {2}/{1,3,4}, {3}/{1,2,4}, {4}/{1,2,3}, all with EI>0. c. Analysis of complexes. By considering all subsets of system X one can identify its complexes and rank them by the respective values of Φ – the value of EI for their minimum information bipartition. Assuming that other elements in X are disconnected, it is easy to see that Φ>0 for subset {3,4} and {1,2}, but Φ = 0 for subsets {1,3}, {1,4}, {2,3}, {2,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, and {1,2,3,4}. Subsets {3,4} and {1,2} are not part of a larger subset having higher Φ, and therefore they constitute complexes. This is indicated schematically by having them encircled by a grey oval (darker grey indicates higher Φ). Methodological note. In order to identify complexes and their Φ(S) for systems with many different connection patterns, each system X was implemented as a stationary multidimensional Gaussian process such that values for effective information could be obtained analytically (details in [8]). Briefly, in order to identify complexes and their Φ(S) for systems with many different connection patterns, we implemented numerous model systems X composed of n neural elements with connections CONij specified by a connection matrix CON(X) (no self-connections). In order to compare different architectures, CON(X) was normalized so that the absolute value of the sum of the afferent synaptic weights per element corresponded to a constant value w<1 (here w = 0.5). If the system's dynamics corresponds to a multivariate Gaussian random process, its covariance matrix COV(X) can be derived analytically. As in previous work, we consider the vector X of random variables that represents the activity of the elements of X, subject to independent Gaussian noise R of magnitude c. We have that, when the elements settle under stationary conditions, X = X * CON(X) + cR. By defining Q = (1-CON(X))-1 and averaging over the states produced by successive values of R, we obtain the covariance matrix COV(X) = <X*X> = <Qt * Rt * R * Q> = Qt * Q, where the superscript t refers to the transpose. Under Gaussian assumptions, all deviations from independence among the two complementary parts A and B of a subset S of X are expressed by the covariances among the respective elements. Given these covariances, values for the individual entropies H(A) and H(B), as well as for the joint entropy of the subset H(S) = H(AB) can be obtained as, for example, H(A) = (1/2)ln [(2π e)n|COV(A)|], where |•| denotes the determinant. The mutual information between A and B is then given by MI(A;B) = H(A) + H(B) - H(AB). Note that MI(A:B) is symmetric and positive. To obtain the effective information between A and B within model systems, independent noise sources in A are enforced by setting to zero strength the connections within A and afferent to A. Then the covariance matrix for A is equal to the identity matrix (given independent Gaussian noise), and any statistical dependence between A and B must be due to the causal effects of A on B, mediated by the efferent connections of A. Moreover, all possible outputs from A that could affect B are evaluated. Under these conditions, EI(A→B) = MI(AHmax;B). The independent Gaussian noise R applied to A is multiplied by cp, the perturbation coefficient, while the independent Gaussian noise applied to the rest of the system is given by ci, the intrinsic noise coefficient. Here cp = 1 and ci = 0.00001 in order to emphasize the role of the connectivity and minimize that of noise. To identify complexes and obtain their capacity for information integration, one considers every subset S of X composed of k elements, with k = 2,..., n. For each subset S, we consider all bipartitions and calculate EI(A B) for each of them. We find the minimum information bipartition MIB(S), the bipartition for which the normalized effective information reaches a minimum, and the corresponding value of Φ(S). We then find the complexes of X as those subsets S with Φ>0 that are not included within a subset having higher Φ and rank them based on their Φ(S) value. The complex with the maximum value of Φ(S) is the main complex. MATLAB functions used for calculating effective information and complexes are at webcite.

Tononi BMC Neuroscience 2004 5:42   doi:10.1186/1471-2202-5-42
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