Synchronization of neuron population subject to steady DC electric field induced by magnetic stimulation

Abstract

Electric fields, which are ubiquitous in the context of neurons, are induced either by external electromagnetic fields or by endogenous electric activities. Clinical evidences point out that magnetic stimulation can induce an electric field that modulates rhythmic activity of special brain tissue, which are associated with most brain functions, including normal and pathological physiological mechanisms. Recently, the studies about the relationship between clinical treatment for psychiatric disorders and magnetic stimulation have been investigated extensively. However, further development of these techniques is limited due to the lack of understanding of the underlying mechanisms supporting the interaction between the electric field induced by magnetic stimulus and brain tissue. In this paper, the effects of steady DC electric field induced by magnetic stimulation on the coherence of an interneuronal network are investigated. Different behaviors have been observed in the network with different topologies (i.e., random and small-world network, modular network). It is found that the coherence displays a peak or a plateau when the induced electric field varies between the parameter range we defined. The coherence of the neuronal systems depends extensively on the network structure and parameters. All these parameters play a key role in determining the range for the induced electric field to synchronize network activities. The presented results could have important implications for the scientific theoretical studies regarding the effects of magnetic stimulation on human brain.

Keywords: Magnetic stimulation; Steady DC electric field; Synchronization; Topology.

Fig. 1

Bifurcation diagram of the single model versus induced electric field. It shows the single neuron from rest to tonic firing and then decays to rest versus the induced electric field V_E. Lines and circles indicate fixed points and periodic solutions, respectively. Thick solid lines indicate stable fixed points; thin lines are unstable fixed points. Solid circles are stable periodic solutions and open circles are unstable periodics. When V_E varies between [−28 −2] mV, the cell can fire spikes

Fig. 2

The action potential of the single model with different values of V_E. The fire rate increases with V_E in (a–e). But when V_E < −28.5 mV, the voltage decays to rest in (f)

Fig. 3

Example of considered random network topologies. Given 100 isolated nodes, one connects every pair of nodes with probability p_r, a 0.01, b 0.1

Fig. 4

The dependence of coherence coefficient K on the induced electric field for different size of the random network. The parameters is p_r = 0.5. This figure is the average result of ten trials

Fig. 5

The dependence of coherence coefficient K on the induced electric field for different values of probability p_r of connections in the random network. The parameters are N = 100 in (a), N = 200 in (b). This figure is the average result of ten trials

Fig. 6

The dependence of coherence coefficient K on the induced electric field for different values of maximal synaptic conductance g_syn each neuron has in the random network. The parameters are N = 100 in (a), N = 200 in (b), p_r = 0.5. This figure is the average result of ten trials

Fig. 7

The dependence of coherence coefficient K on the induced electric field for different fraction PN of elements in the neuron population receiving stimulation. The parameters are N = 100 in (a), N = 200 in (b), p_r = 0.5. This figure is the average result of ten trials

Fig. 8

Example of considered small-world network topologies. Given 20 isolated nodes. a Regular ring characterized by p = 0. Each node is connected to its k = 4 nearest neighbors. b Realization of small-world topology via random rewiring of a certain fraction p = 0.1 of links

Fig. 9

The dependence of coherence coefficient K on the induced electric field for different size of the small-world network. The parameters is p = 0.5, k = 4. This figure is the average result of ten trials

Fig. 10

The dependence of coherence coefficient K on the induced electric field for different number of nearest neighbors in the small-world network. The parameters is p = 0.5, N = 100. This figure is the average result of ten trials

Fig. 11

The dependence of coherence coefficient K on the induced electric field for different values of rewiring probability in the small-world network. The parameters is k = 4, N = 100. This figure is the average result of ten trials

Fig. 12

The dependence of coherence coefficient K on the induced electric field for different values of maximal synaptic conductance g_syn each neuron has in the small-world network. The parameters are N = 100, k = 4, p = 0.5. This figure is the average result of ten trials

Fig. 13

The dependence of coherence coefficient K on the induced electric field for different fraction PN of elements in the neuron population receiving stimulation in the small-world network. The parameters are N = 100, k = 4, p = 0.5. This figure is the average result of ten trials

Fig. 14

Schematic presentation of the considered modular network architecture. The modular network consists of M = 4 small-world subnetworks, each containing NI = 20 neurons with rewiring probability p = 0.05. The subnetwork 1 is taken as a central module connected to other modules. There are three connections among neurons between the central module and any other module

Fig. 15

The dependence of coherence coefficient K on the induced electric field for different values of a inter-synaptic conductance g_intra between the subnetworks, b intra-synaptic conductance g_intra in the subnetworks within the modular network. The parameters are M = 4, NI = 100, k = 4, p = 0.5, P_inter = 0.1, PN = 0.5. This figure is the average result of ten trials

Fig. 16

The dependence of coherence coefficient K on the induced electric field for different number of nearest neighbors k in the subnetwork within the modular network. The parameters are M = 4, NI = 100, g_intra = 0.02, g_intra = 0.01, p = 0.5, P_intra = 0.1, PN = 0.5. This figure is the average result of ten trials

Fig. 17

The dependence of coherence coefficient K on the induced electric field for different values of rewiring probability p in the subnetwork within the modular network. The parameters are M = 4, NI = 100, g_intra = 0.02, g_intra = 0.01, k = 4, P_intra = 0.1, PN = 0.5. This figure is the average result of ten trials

Fig. 18

The dependence of coherence coefficient K on the induced electric field for different values of inter-connection probability P_intra between the subnetworks within the modular network. The parameters are M = 4, NI = 100, g_intra = 0.02, g_intra = 0.005 in (a) and g_intra = 0.010 in (b), k = 4, p = 0.5, PN = 0.5. This figure is the average result of ten trials

Fig. 19

The dependence of coherence coefficient K on the induced electric field for different fraction PN of elements in the neuron population receiving the external drive in the modular network. The parameters are M = 4, NI = 100, g_intra = 0.02, g_intra = 0.01, k = 4, P_intra = 0.1. This figure is the average result of ten trials

Fig. 20

The dependence of coherence coefficient K on the induced electric field for different number M of the subnetworks within the modular network. The parameters are NI = 100, g_intra = 0.02, g_intra = 0.01, k = 4, p = 0.5, PN = 0.5. This figure is the average result of ten trials