Synaptic changes modulate spontaneous transitions between tonic and bursting neural activities in coupled Hindmarsh-Rose neurons

Experimentally, certain cells in the brain exhibit a spike-burst activity with burst synchronization at transition to and during sleep (or drowsiness), while they demonstrate a desynchronized tonic activity in the waking state. We herein investigated the neural activities and their transitions by using a model of coupled Hindmarsh-Rose neurons in an Erdős-Rényi random network. By tuning synaptic strength, spontaneous transitions between tonic and bursting neural activities can be realized. With excitatory chemical synapses or electrical synapses, slow-wave activity (SWA) similar to that observed during sleep can appear, as a result of synchronized bursting activities. SWA cannot appear in a network that is dominated by inhibitory chemical synapses, because neurons exhibit desynchronized bursting activities. Moreover, we found that the critical synaptic strength related to the transitions of neural activities depends only on the network average degree (i.e., the average number of signals that all the neurons receive). We demonstrated, both numerically and analytically, that the critical synaptic strength and the network average degree obey a power-law relation with an exponent of $-1$. Our study provides a possible dynamical network mechanism of the transitions between tonic and bursting neural activities for the wakefulness-sleep cycle, and of the SWA during sleep. Further interesting and challenging investigations are briefly discussed as well.

Figure 1

The diagram of ISIs versus external current $I_{\text{ext}}$ for a single HR neuron. The insets show the time series of bursting (at $I_{\text{ext}}=2.5$) and tonic (at $I_{\text{ext}}=3.6$) activities of the membrane potential $x$.

Figure 2

The two spontaneous transitions between the tonic and the bursting neural activities in an ER random network with size $N=100$ and excitatory chemical connection probability $p=0.1$. (a) The change of synaptic strength $g$. (b) The dot-raster plot showing the firing pattern. (c) The average potential signal $\overline{x}$ of the whole network. (d) The membrane potential $x_i$, (e) the slow current $z_i$, and (f) the sum of external current $I_{\text{ext}}$ and synaptic coupling current $I_{i\left(\text{syn}\right)}$ for a randomly chosen neuron $i$. The two vertical red dashed lines illustrate the critical times of the two transitions.

Figure 3

The diagram of ISIs versus synaptic strength $g$ for a randomly chosen neuron $i$ in the ER random network with a scale of $N=100$ and an excitatory chemical connection probability $p=0.1$. The insets demonstrate the time series of the tonic (at $g=0.008$) and the bursting (at $g=0.05$ and $g=0.1$) activities of the membrane potential $x_i$ (top panels) for the chosen neuron $i$ as well as the average potential signal $\overline{x}$ (bottom panels) for the whole network. The vertical red dashed line illustrates the critical $g^{*}$ producing the transition between the tonic and the bursting activities.

Figure 4

The same as in Fig. 2, but for an electrically coupled network.

Figure 5

The same as in Fig. 3, but for an electrically coupled network. The insets are given at $g=0.003, g=0.01$, and $g=0.04$.

Figure 6

The same as in Fig. 2, but for inhibitory chemical synaptic connections.

Figure 7

The same as in Fig. 3, but for inhibitory chemical synaptic connections. The insets are given at $g=0.003, g=0.015$, and $g=0.045$.

Figure 8

The same as in Fig. 2, but for a chemically coupled network including both excitatory and inhibitory synapses at a ratio of 4:1.

Figure 9

The same as in Fig. 2, but for a chemically coupled network including both excitatory and inhibitory synapses at a ratio of 1:4.

Figure 10

The critical $g^{*}$ as a function of $N$ for different connection probabilities $p=0.3,0.5,0.7,1.0$ (i.e., globally coupled network) with chemical (a) or electrical (b) synapses. Data are averaged over ten realizations of the networks with random initial conditions. For comparison, the solid lines indicating the power-law property with an exponent of $-1$ are shown in (a) and (b).

Figure 11

The same as in Fig. 10, but as a function of the network average degree $\overline{k}$.

Figure 12

The diagram of ISIs versus $c$ at $\omega =0.3$ for excitatory (a) or inhibitory (b) chemical synapses, and electrical synapses (c) in the presence of synaptic oscillating currents Eqs. (6, 7, 8), respectively. The vertical red dashed lines illustrate the critical $c^{*}$ producing the transition between tonic and bursting activities.

Figure 13

The diagram of $c^{*}$ versus $\omega$ for different synapses in the presence of corresponding synaptic oscillating currents. Data are averaged over ten realizations with random initial conditions.