Analyzing synchronized clusters in neuron networks

Abstract

The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. Common approaches to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to two examples with different size and features (the directed network of the macaque cerebral cortex and the swim central pattern generator of a mollusc) provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some modeling simplifications in one of the examples, the obtained results are in good agreement with previously reported biological data.

Figure 1

Example. (A) Network with $N=11$ nodes, $L=2$ kinds of connection, undelayed ($k=1$) or with delay ${\delta }_2$ ($k=2$), and $Q=4$ clusters ($C_1=\{1,2,3,4\},C_2=\{5,6\},C_3=\{7,8,9\},C_4=\{10,11\}$). All connections are bi-directional and with weight 1, with the exception of the thick connections (between nodes 5-7, 1-5, 6-9, 3-6), which have weight 2. The connection between nodes 5 and 6 has the delay ${\delta }_2$. (B) Quotient network corresponding to (A). (C) Structure of the corresponding matrices T and $B^1$, illustrating their relation with the clusters. Network coloring (with a larger number of clusters) after the breaking of the red cluster if its loss of stability is due to the MLEs corresponding to either (D) the multi-color sub-block or (E) the red sub-block of matrix $B^1$.

Figure 2

(A) Swim CPG of the Dendronotus iris nudibranch mollusc. Lines terminating in filled circles indicate inhibitory chemical synapses ($k=1$). Triangles indicate fast excitatory chemical synapses ($k=2$). Resistor symbols indicate electrical (gap junction) connections ($k=3$). Neurons 1-3 are located in the left half of the mollusc brain, neurons 4-6 in the right half of the brain. (B) Structure of the matrices T, $B^1$, $B^2$, and $B^3$ for the swim CPG network. The gray blocks correspond to 0 entries.

Figure 3

Time plots of the membrane voltages $V_i(t)$ for the swim CPG in normal conditions (A) and in conditions emulating (by setting ${\sigma }^1=0$ and ${\sigma }^2=0$) a bath application of curare (B). Cluster $C_1$ (top panel), $C_2$ (middle panel), $C_3$ (bottom panel). Blue lines: $V_i(t)$ for $i=1,2,3$. Red lines: $V_i(t)$ for $i=4,5,6$. (C) Two-parameter map of the stable clusters for the swim CPG. Green region: all clusters ($C_1,C_2,C_3$) are stable (A). Red region: $C_1,C_2,C_3$ lose their stability (B). Yellow region: bi-stability transition zone.

Figure 4

(A) Macaque cortical connectivity network: $N=29$ nodes, $M=2$ node models, $L=2$ synapse models. Trivial clusters are black. Nodes of the same (non-black) color belong to the same cluster: C₁ (green), C₂ (red), C₃ (blue). (B) ECs of the macaque cortical network.

Figure 5

Time plots $V_i(t)$ for different values of ${\delta }_2$ (5 ms (A), 15 ms (B)) for cluster $C_3$. (C) MLE ${\mathrm{\Lambda }}_{C_q}$ of each cluster $C_q$ ($q=2,3$) vs. coupling delay ${\delta }_2$, for the macaque cortical connectivity network. Horizontal dashed lines: edge of stability. Vertical dotted lines: ${\delta }_2$ values corresponding to the time plots in panels A and B.

Figure 6

Time responses (firing rates) to a pulse-shaped input to area V1.