Structural condition for oscillations: odd inhibitory cycle rule and its illustrations.

a: Examples of oscillating motifs and non-oscillating motifs in Wilson-Cowan model. Motifs that cannot oscillate show features of Winner-take-all: the winner will inhibit other nodes with a high activity level. Inversely, the oscillatory ones all show features of winner-less competition, which may contribute to oscillation. b: The odd inhibitory cycle rule for oscillation prediction with the sign condition of a network. c: Illustrations of oscillation in complex networks. Based on the odd inhibitory cycle rule, Network I can’t oscillate, while Network II could oscillate by calculating the sums of their motifs. The red or black arrows indicate inhibition or excitation, respectively. Hollow nodes and solid nodes represent excitatory and inhibitory nodes, respectively.

The intuitive explanations of Theorem 1 and 2.

a: A visual representation of why directed cycles are important in network oscillation. By rearranging all nodes, any network without directed cycles can be seen as a feed-forward network which will make the system reach a stable fixed point. b: An intuitive explanation of the odd inhibitory cycle rule by showing the activities of two 6-node-loops. Odd inhibitory connections (bottom) can help the system oscillate, while even inhibitory connections has the opposite effect.

Influence of network properties on the oscillation frequency in motifs III and EII with Wilson-Cowan model.

a: The changed network parameters are shown in the table. Red (green) connections are inhibitory (excitatory) and black arrows are the external inputs. b-e: We systematically varied the synaptic delay time b, synaptic weights c, external input d, and self-connection e. These parameters were varied simultaneously for all the synapses i.e. in each simulation all synapses were homogeneous. Green, orange, red and turquoise respectively show the effect of synaptic delay, synaptic strength, external input and self-inhibition. See the figure supplement 3-1 and figure supplement 3-3 for more detailed results about III and EI network motifs.

Figure 3—figure supplement 1. Influence of III network properties on the oscillation frequency in Wilson-Cowan model. The controlled properties in motif III, including delay time a, synaptic weights b, external input c, and self-connection d, are denoted successively by green, orange, red and blue. We controlled two factors once at a time to observe the reaction of oscillation frequency with sketch maps on the right as conclusions.

Figure 3—figure supplement 2. Effect of Synaptic Delays on Motifs with Even Inhibition in the Wilson-Cowan Model. The synaptic delays are varied from 2ms to 10ms in motifs II, EEE, and EII, with corresponding external inputs to nodes being [4, 4], [2.2, 2, 1.8], and [3, 3, 3]

Figure 3—figure supplement 3. Influence of EI network properties on the oscillation frequency in Wilson-Cowan model. The controlled properties in motif EI, including delay time a, synaptic weights b, external input c, and self-connection d, are denoted successively by green, orange, red and blue. We controlled two factors once at a time to observe the reaction of oscillation frequency with sketch maps on the right as conclusions.

Schematic of CBG network model with potential oscillators and the interaction between two oscillators in Wilson-Cowan model.

a: CBG structure with red lines denoting inhibition and green lines denoting excitation, along with five potential oscillators based on the odd inhibitory cycle rule. b: Oscillation in all BG motifs from 2 nodes to 6 nodes based on the odd inhibitory cycle rule. Each grid represents a separate motif. We use different colors to mark potential oscillators in each motif in BG, and each color means an oscillator from panel a. For more details, see figure supplement 4-.1–figure supplement 4-6. c: The reaction of oscillation frequency to different external inputs to D2 and STN in a BG subnetwork. External inputs to Proto and Arky are 1 and 3, respectively. d: Same thing as c but ruining the connection from D2 to Proto. e: Same thing as c but destroying the connections from STN and increasing the input to Proto from 1 to 4.

Figure 4—figure supplement 1. All 2-node-motifs in CBG network. Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif

Proto-GPi-Th-Cortex-D2.

Figure 4—figure supplement 2. All 3-node-motifs in CBG network. Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

Figure 4—figure supplement 3. All 4-node-motifs in CBG network. Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

Figure 4—figure supplement 4. All 5-node-motifs in CBG network. Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex;Blue: motif Proto-Arky-D2;Green: motif Proto-FSN-D2;Purple: motif Proto-GPi-Th-Cortex-D2.

Figure 4—figure supplement 5. All 6-node-motifs in CBG network. Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif

Oscillations in a leaky integrate-and-fire (LIF) spiking neuronal network model of specific BG motifs.

a-b: Average peristimulus time histograms (PSTH) of all neurons in a Proto-FSN-D2 and b Proto-Arky-D2 motifs under Parkinson condition with power spectral density (PSD) at the top right. c: PSTH of Proto and STN in a BG subnetwork with motif Proto-Arky-D2 as the oscillator during different STN inhibition. d: Same thing as c but changing the oscillator from Proto-Arky-D2 to Proto-STN.

Parameters of III network for Fig. 3 and 1

Parameters of EII network for Fig. 3

Parameters for Fig. 4 (Wilson-Cowan model)

Parameters for the EI network in Fig. 3 (Wilson-Cowan model)

Parameters of D2-SPN neurons (LIF model with conductance-based synapses)

Parameters of FSN neurons (LIF model with conductance-based synapses)

Parameters of STN neurons (LIF model with conductance-based synapses)

Parameters of Proto and Arky neurons (LIF model with AdEx)

Synaptic conductance weight and delay parameters in LIF model

Number of connections on each neuron and constant input current for Fig. 5c (LIF model)

Number of connections on each neuron and constant input current for Fig. 5d (LIF model)

Influence of III network properties on the oscillation frequency in Wilson-Cowan model.

The controlled properties in motif III, including delay time a, synaptic weights b, external input c, and self-connection d, are denoted successively by green, orange, red and blue. We controlled two factors once at a time to observe the reaction of oscillation frequency with sketch maps on the right as conclusions.

Effect of Synaptic Delays on Motifs with Even Inhibition in the Wilson-Cowan Model.

The synaptic delays are varied from 2ms to 10ms in motifs II, EEE, and EII, with corresponding external inputs to nodes being [4, 4], [2.2, 2, 1.8], and [3, 3, 3]

Influence of EI network properties on the oscillation frequency in Wilson-Cowan model.

The controlled properties in motif EI, including delay time a, synaptic weights b, external input c, and self-connection d, are denoted successively by green, orange, red and blue. We controlled two factors once at a time to observe the reaction of oscillation frequency with sketch maps on the right as conclusions.

All 2-node-motifs in CBG network.

Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

All 3-node-motifs in CBG network.

Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

All 4-node-motifs in CBG network.

Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

All 5-node-motifs in CBG network.

Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

All 6-node-motifs in CBG network.

Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.

All 7-node-motifs in CBG network.

Potential oscillating motifs behind these subnetworks are marked with different colors. Yellow: motif Proto-STN; Orange: motif STN-GPi-Th-cortex; Blue: motif Proto-Arky-D2; Green: motif Proto-FSN-D2; Purple: motif Proto-GPi-Th-Cortex-D2.