Symmetry breaking in soft clustering decoding of neural codes
© Dimitrov et al; licensee BioMed Central Ltd. 2009
Published: 13 July 2009
We have been able to answer several questions about these bifurcations:.
1. There are N - 1 symmetry breaking bifurcations observed when continuing from the initial solution because there are only N - 1 subgroups in the chain S N → SN-1→...→ S2 → S1.
2. The annealing solutions in Figure 1 all have symmetry S M for some M <N. There exist other suboptimal (and unstable) branches with symmetry S m × S n (m + n = N) that yield mutual information values below the envelope curve depicted in the figure.
3. Symmetry breaking bifurcations are generically pitchfork-like and derivative calculations predict whether the bifurcating branches are subcritical or supercritical, as well as their stability. Symmetry preserving bifurcations are generically saddle nodes.
4. A local solution to the optimization problem does not always bifurcate through a symmetry breaking bifurcation.
5. The bifurcations of solutions dictate the convexity of the curve in Figure 1. In particular, a subcritical bifurcation of solutions at I0 implies that the rate-distortion curve R(I) changes convexity in a neighborhood of I0. This is in contrast to the rate distortion curve in information theory, R(D), for which D(q) is linear in q.
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