Intermittent synchronization in a network of bursting neurons - PubMed (original) (raw)
Intermittent synchronization in a network of bursting neurons
Choongseok Park et al. Chaos. 2011 Sep.
Abstract
Synchronized oscillations in networks of inhibitory and excitatory coupled bursting neurons are common in a variety of neural systems from central pattern generators to human brain circuits. One example of the latter is the subcortical network of the basal ganglia, formed by excitatory and inhibitory bursters of the subthalamic nucleus and globus pallidus, involved in motor control and affected in Parkinson's disease. Recent experiments have demonstrated the intermittent nature of the phase-locking of neural activity in this network. Here, we explore one potential mechanism to explain the intermittent phase-locking in a network. We simplify the network to obtain a model of two inhibitory coupled elements and explore its dynamics. We used geometric analysis and singular perturbation methods for dynamical systems to reduce the full model to a simpler set of equations. Mathematical analysis was completed using three slow variables with two different time scales. Intermittently, synchronous oscillations are generated by overlapped spiking which crucially depends on the geometry of the slow phase plane and the interplay between slow variables as well as the strength of synapses. Two slow variables are responsible for the generation of activity patterns with overlapped spiking, and the other slower variable enhances the robustness of an irregular and intermittent activity pattern. While the analyzed network and the explored mechanism of intermittent synchrony appear to be quite generic, the results of this analysis can be used to trace particular values of biophysical parameters (synaptic strength and parameters of calcium dynamics), which are known to be impacted in Parkinson's disease.
Figures
Figure 1
Network architecture. (a) Reciprocally connected large STN-GPe network. Arrow indicates excitatory connection and circle indicates inhibitory connection. (b) Reduced network of two inhibitory cells with self-inhibition.
Figure 2
(a) Averaged coherence over 0–100 Hz band (black) and 10–30 Hz band (gray) between voltages of two cells. Coupling strength _g_syn varies from 0.1 to 2 with 0.1 stepsize. (b) Phase-locking index γ in dependence on _g_syn. Different lines denote different strength of self-coupling; default value is 30% (black solid) and two other strengths, 25% (dotted) and 35% (gray) are also shown for comparison. (c) Activity patterns with irregular sequence of burstings with overlapped spikes when _g_syn=0.9.
Figure 3
Return map r n+1 vs. r n for different values of the synaptic strength parameter _g_syn (a) presents five different values of _g_syn: 0.86(square), 0.88(circle), 0.94(triangle), 0.96(diamond), and 0.92(gray dots). (b) More complex dynamics observed for _g_syn=0.9. (c) Maximal Lyapunov exponents (MLEs) over a range of refined _g_syn values between 0.9 and 0.92. When _g_syn lies between 0.9 and 0.91, MLE values are significant as compared with other range of _g_syn values. Over this range of _g_syn values, chaotic activity patterns may be robust.
Figure 4
Irregular partially-synchronous dynamics of reduced model when g_syn=0.9. (a) An example of escaping. Upper panel shows voltage profiles and lower panel shows corresponding phases for two neurons. (b) Return map for phases φ_i+1 vs. φ_i_. (c) Histrogram of durations of desynchronization events. Black bars come from the experimental data.
Figure 5
(a) STN cell activity patterns when _g_syn=0.9. Upper panel shows voltage profile and lower panel shows slow variables r (black solid), σ (lower gray solid), and [_Ca_] (black dotted). Refer to Eqs. 2, 3, 4. The time course of [_Ca_] is also plotted with the scale on the right vertical axis (upper gray solid). (b) Bifurcation diagram of the fast subsystem for r = 1 and [_Ca_] = 0.7. There is a saddle-node bifurcation point (SN) at σ = σSN and also a subcritical Hopf bifurcation point on the upper branch where an unstable periodic orbit begins to be turning to a stable periodic orbit. This stable periodic orbit becomes a saddle-node homoclinic orbit when σ = σSN. Stable (unstable) fixed points and limit cycles are in thick black (thin gray).
Figure 6
The frequency of firing in dependence on the slow variables σ and r. (a) Σ-curve (gray line in the (σ, r) plane) divides the space of the slow variables (σ, r) into silent and sustained spiking regions. Over the sustained spiking region, the curves of frequencies are plotted for various r (0.25, 0.3, 0.35, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1, left to right). (b) Another view of frequency curves from the part (a). For larger values of r, the frequency of the periodic solution decreases almost linearly as σ decreases and then sharply decreases near Σ. For r ≥ 0.5, the frequency curves are almost identical.
Figure 7
Two-parameter bifurcation diagram with projection of 2-spike out-of-phase bursting solution. The close-to-vertical curve in the middle of the figure is the Σ-curve shown in Fig. 6 when [_Ca_] = 0.7. The moment when active cell fires its last spike is denoted by lower circle and upper circle denotes (σ, r) of silent cell at that moment. Open circle denotes the moment that silent cell is released from inhibition. The moment when the silent cell fires its first spike is denoted by upper square and lower square denotes (σ, r) of active cell at that moment. _T_1 is the time needed for silent cell to fire its first spike after it is released from inhibition (from open circle to upper square) and _T_2 is inter-spike interval between first and second spike. Black line is the trajectory of active cell from its first spike until the moment that it gets inhibition to leave the spiking region. The counterpart of silent cell is denoted by gray trajectory.
Figure 8
Spike firing for different values of r and [_Ca_]. (a) The number of spikes that a cell fires depends on the level of r when the cell is released from inhibition. The vertical line (σ = 0.927) is divided into three intervals according to number of spikes, 2, 3, and 4. Boundaries between these intervals are marked by two dots (r = 0.45 and 0.58). Three exemplary cases for each number of spikes are presented for r is 0.4 (2 spikes, dotted), 0.5 (3 spikes, black solid), and 0.6 (4 spikes, gray solid). (b) Plot of _T_1 (left black solid), _T_2 (left gray solid), _T_3 (right gray solid) and _T_4 (right black solid) as a function of r. The dotted lines are T k curves for k = 1, 2, 3, and 4 when [_Ca_] is slow variable. (C) Σ-curves for several [_Ca_] levels from 0.4 to 0.8 with stepsize 0.1 (left to right). Values of r and σ are also plotted when a cell fires its first spike for various initial values of r, from 0.35 to 0.7 with the stepsize 0.05 (from bottom to top). Synaptic strength _g_syn=0.9.
Figure 9
An example of “escaping” for _g_syn=0.9; one cell fires three spikes and during the firing of its last spike, the other cell also fires. (a) Voltage traces of both cells (only last two spikes are shown in this figure, black line is escaping cell). (b) The corresponding trajectory in the phase space of slow variables. Black dots (squares) correspond to time t = 0 (t = 40) in part (a). (c) _T_1 (solid) and _T_3 (dotted) curves as a function of r when [_Ca_] is variable. Initial conditions for [_Ca_] are 0.66 (black) and 0.7 (gray).
Figure 10
r* and r f values for the range of _g_syn and different values of [_Ca_]. The lower four circles and upper four squares are numerically computed r*’s for 2-spike and 3-spike regular bursting solutions. The middle three circles and squares are obtained by quadratic extrapolation. Numerically computed r _f_’s for different [_Ca_] levels are also shown (gray stars and lines, [_Ca_] level increases from 0.6, bottom on the right, to 0.7, top on the right, with the stepsize of 0.02).
Figure 11
Escaping revisited. (a) An escaping example in the (r, [_Ca_]) plane. Close to vertical lines divide the plane according to the number of spikes when a cell is released from inhibition (Fig. 8). Left to the dotted line is 1-spike region and right 2-spike region. Similarly, right to the dashed line is 3-spike region. Solid line is the line of r f in the 3-spike region. Four circles between middle and right vertical lines denote the values of r and [_Ca_] when silent cell is lastly inhibited in a regular 2-spike bursting solution. From bottom to top, _g_syn values are from 0.86 to 0.89. Square at the top of this array of circles denotes the extrapolated values of r and [_Ca_] when _g_syn is 0.9. Lower trajectory is active cell and upper trajectory is silent cell and triangles denote starting points of each trajectory. Stars denote the moments when cells get inhibition either from itself or from other cell. (b) Three gray solid lines are T k curves using the calcium level when the active cell is lastly inhibited. Dots on these curves denote _T_2 and _T_3 of active cell computed from the value of r when the active cell is released. Black curve is _T_1 of silent cell when it gets second inhibition and square denotes actual _T_1 value at that moment.
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