Computational model of electrically coupled, intrinsically distinct pacemaker neurons

Abstract

Electrical coupling between neurons with similar properties is often studied. Nonetheless, the role of electrical coupling between neurons with widely different intrinsic properties also occurs, but is less well understood. Inspired by the pacemaker group of the crustacean pyloric network, we developed a multicompartment, conductance-based model of a small network of intrinsically distinct, electrically coupled neurons. In the pyloric network, a small intrinsically bursting neuron, through gap junctions, drives 2 larger, tonically spiking neurons to reliably burst in-phase with it. Each model neuron has 2 compartments, one responsible for spike generation and the other for producing a slow, large-amplitude oscillation. We illustrate how these compartments interact and determine the dynamics of the model neurons. Our model captures the dynamic oscillation range measured from the isolated and coupled biological neurons. At the network level, we explore the range of coupling strengths for which synchronous bursting oscillations are possible. The spatial segregation of ionic currents significantly enhances the ability of the 2 neurons to burst synchronously, and the oscillation range of the model pacemaker network depends not only on the strength of the electrical synapse but also on the identity of the neuron receiving inputs. We also compare the activity of the electrically coupled, distinct neurons with that of a network of coupled identical bursting neurons. For small to moderate coupling strengths, the network of identical elements, when receiving asymmetrical inputs, can have a smaller dynamic range of oscillation than that of its constituent neurons in isolation.

FIG. 1

Activity of the anterior burster (AB) and pyloric dilator (PD) neurons in response to current injection with intact modulatory inputs to the stomatogastric ganglion (STG). Gray background shows 0 current injection, + denotes positive current injection, and − denotes negative current injection. A: voltage traces of the biological (left) AB (red) and PD (blue) neurons show in-phase bursting oscillations that become faster with increasing current injection in the AB neuron. This behavior is mimicked by the model neurons (right). I_ext in nA from top: Biological, 5, 3, 0.5, 0, −8, −8.5; Model, 5, 1.1, 0.8, 0, −0.22, −0.3. Minimum V_m in mV from top: Biological AB, 36, −41, −49, −51, −85, −86; Biological PD, −41, −42, −45, −46, −53, −54; Model AB, −45.2, −47.5, −48, −53, −52.4, −51; Model PD, −46, −46.5, −48.1, −53, −52.4, −51. Model pacemaker has a frequency range of 0.22 to 1.7 Hz. B: voltage traces of the synaptically isolated biological (left) and model (right) AB neurons show bursting oscillations that become faster and smaller in amplitude with injected current. I_ext in nA from top: Biological, 33, 12, 6, 0, −3, −5; Model: 8, 1.0, 0.3, 0, −0.19, −0.27. Minimum V_m of the AB neuron in mV from top: Biological, 27, −17, −36, −60, −66, −76; Model, −44.6, −51, −54.8, −58.4, −59.2, −52.75. Model AB neuron has a frequency range of 0.24 to 3.4 Hz. C: voltage traces from the isolated biological (left) and model (right) PD neurons show tonic spiking activity that increases in frequency with current injection. I_ext in nA from top: Biological, 15, 0, −1; Model: 12, 0, −0.3. Minimum V_m of the PD neuron in mV from top: Biological, −27, −50, −53; Model, −44, −46.5, −53. In the model, current injection runs were started at 0 nA with subsequent steps of 0.1 or 0.2 nA that lasted 20 to 30 s ≤8 nA (in A), 1.5 nA (in B), or 12 nA (in C). Current was then reset to 0 nA and subsequently decreased to −0.3 nA in steps of −0.01 nA or −0.02 nA.

FIG. 2

Activity of the AB and PD neurons in response to current injection in the AB neuron in the absence of modulatory input. Biological AB (red) and PD (blue) neurons are quiescent with no current injections (gray background; left traces). Positive DC current injection (+) in the biological AB neuron evokes in-phase bursting oscillations that become faster with increasing current levels (left). This behavior is reproduced by the model neurons (right). I_ext in nA from top: Biological, 10, 6, 1, 0; Model, 1, 0.6, 0.2, 0. Minimum V_m in mV from top: Biological AB neuron, −12, −35, −66, −77; Biological PD neuron, −51, −52, −55, −56; Model AB neuron, −45.3, −45.8, −48.2, −49.7; Model PD neuron, −47.7, −47.7, −48.5, −49.8. In the model, current injection runs were started at 0 nA, with subsequent steps of 0.2 to ≤2 nA. Each run was 30 s.

FIG. 3

Membrane potential waveform of the AB neuron is changed by electrical coupling to the PD neuron. A: intracellular recordings show the waveforms of an AB neuron when electrically coupled (thin trace) to the PD neuron and when isolated (thick trace). On isolation, the average spike amplitude increased and the average spike number per burst decreased. Minimum V_m (in mV): −61.3 for coupled and −62.9 for isolated AB neuron. B: a comparison of the waveforms in the model AB neuron when it is coupled to the PD neuron (thin trace) and when isolated (thick trace) shows that the isolated AB neuron waveform has a larger amplitude and shorter period, as in the biological neuron. Spike number per burst also decreased. Minimum V_m (in mV): −53.5 for coupled and −58.4 for isolated AB neuron.

FIG. 4

Compartmentalization of the model neurons. A: schematic representation of the distribution of intrinsic currents in the model neurons. Ionic currents responsible for action potential generation were placed in the “A” (axon) compartment. Ionic currents underlying the generation of slow oscillations were placed in the S/N (soma-primary neurite) compartment. Presence of anterior inputs was modeled with a voltage-gated inward current I_proc in the model AB neuron, and with larger calcium currents I_CaS and I_CaT in the model PD neuron. In B, C, and D, the top voltage traces correspond to the AB neuron and the bottom voltage traces to the PD neuron; the left traces show the membrane potential of the S/N compartments, whereas those on the right show the A compartments. B: activity of each compartment is shown in isolation, with parameters as described in Table 2. Lowest voltage values are for the AB neuron: −63.5 mV (S/N), −60 mV (A); for the PD neuron: −73 mV (S/N), −57.5 mV (A). C: each S/N compartment shown in B is joined to the corresponding A compartment, with axial conductances as in Table 2. Resulting input resistances were 14.8 MΩ (AB) and 6.9 MΩ (PD). Lowest voltage values are for the AB neuron: −58.4 mV (S/N), −74 mV (A); for the PD neuron: −46.5 mV (S/N), −72.5 mV (A). D: 2 S/N compartments shown in C are electrically coupled to simulate a gap junction with maximal conductance as in Table 2. Two model neurons now burst in-phase. Lowest voltage values are for the AB neuron: −53.5 mV (S/N), −74 mV (A); for the PD neuron: −53.5 mV (S/N), −73 mV (A).

FIG. 5

Effects of varying the axial and gap-junctional conductances. Gray boxes show the activity of the “reference” model. A: behavior of the S/N compartment of the isolated model AB neuron as the axial coupling between the S/N and A compartments is increased. Values (in μS) shown to the right are the maximal conductance of the axial current. Most hyperpolarized voltages in each trace are, from top (in mV): −61, −58.5, −58.4, −59.5, and −69. B: behavior of the S/N compartment of the isolated model PD neuron as the axial coupling between the S/N and A compartments is increased. Values (in μS) shown to the right are the maximal conductance of the axial current; note that there is bistability at 0.8 μS. There was no change in the behavior of the neurons when couplings larger than those shown were used. Most hyperpolarized voltages are (from top to bottom in mV): −71, −67.8, −46, −46.5, −48.5, −52, −55, and −64. C: behavior of the pacemaker AB–PD network as the gap-junctional conductance is increased. Red traces correspond to the AB neuron, the blue traces to the PD neuron. Traces are superimposed on the right to allow a direct comparison of the waveforms. Values (in μS) shown to the right are the gap junction conductances. Most hyperpolarized voltages are (from top to bottom in mV): −52.6, −54.7, −53.5, −57, and −58. In all cases, the simulations were started from the “reference” AB–PD model (Table 2), and the conductance of interest was changed in increments or decrements of 0.05 μS, with each run using as initial conditions the last point of the previous run.

FIG. 6

Comparison of the behavior of different AB–PD networks constructed from one- and 2-compartment model neurons. A: each column shows examples of outputs from a given network as the maximal conductance of the gap junction is increased from top to bottom. Red voltage traces correspond to the model AB neuron and blue traces to the model PD neuron. Diagram on top of each column describes the type of model neuron used. Two-compartment models are shown with a thin line connecting the S/N and A compartments; one-compartment model neurons are shown with whole region between the 2 compartments shaded. Each of the 2 columns in Cases I–IV is built with the same AB neuron but 2 different PD neurons. Gray box shows the activity of the “reference” AB–PD network. Isolated neurons differ from the maximal conductances shown in Table 2 as follows. Case I AB: no difference; Case II AB g_CaT = 42 μS and g_axial = 100 μS; Case III AB: g_axial = 100 μS; Case IV AB: no difference. Cases I, II, and III model PD neurons: g_Na = 2500 μS in the A compartment and g_axial = 100 μS. Cases I, II, and III model PD neurons: g_Na = 2500 μS in the A compartment, g_KCa = 0 and g_axial = 500 μS. Case IV PD: no difference; Case IV PD: g_KCa = 0. Values for the gap-junctional conductances are (from top in μS): Case I AB–PD: 0, 0.4, 1.0, 1.6, 100. Case I AB–PD: 0, 0.1, 1.6, 2, 100. Case II AB–PD: 0, 0.2, 0.75, 3, 100. Case II AB–PD: 0, 0.2, 1.5, 3, 100. Case III AB–PD: 0.4, 1.2, 2,100. Case III AB–PD: 0, 0.2, 0.8, 1.5, 100. Case IV: 0, 0.1, 0.75, 3, 100. Most hyperpolarized voltage values for the traces are (from top in mV). Case I AB–PD, AB: −58.2, −51, −52, −53, −69; PD: −71.2, −71, −70, −68, −69. Case I AB–PD, AB: −58.2, −50.2, −54.5, −68, −62.7, −68; PD: −73, −73, −70, −73.2, −68. Case II AB–PD, AB: −58.4, −45.8, −49.4, −63.6, −69; PD: −71.2, −71.2, −70, −71.5, −69. Case II AB–PD, AB: −58.4, −46.4, −51.6, −63.2, −69; PD: −72, −71.8, −69, −71, −69. Case III AB–PD, AB: −68, −51.6, −53.5, −65.2, −68.8; PD: −71.2, −71, −69.4, −72, −68.8. Case III AB–PD, AB: −68, −58.2, −49.2, −65.5, −68.8; PD: −72, −71.5, −70, −71.2, −68.8. Case IV AB–PD, AB: −58.2, −54.5, −52.9, −53.7, −59.9; PD: −71.2, −51.2, −53, −56.7, −59.9. Case IV AB–PD, AB: −58.2, −49.8, −51.5, −55.8, −59.5; PD: −72, −47, −51.6, −55.8, −59.5. B and C: ratio of the AB neuron burst amplitude when coupled to the PD neuron to the burst amplitude of the isolated AB neuron shown as a function of the maximal gap-junctional conductance. A zero ratio corresponds to a coupled AB neuron that produces tonic spikes instead of bursting oscillations. B: plots of the burst amplitude ratios are shown for both the AB–PD network (B) and the AB–PD network (C) in Cases I–IV.

FIG. 7

Oscillation period range of a model AB–AB network and the model AB–PD network as a function of the gap-junctional strength (in μS). Burst period range was obtained by DC current injection in the AB neuron (A, B) or the PD neuron (C), as depicted in the inset schematic network diagrams. Dashed curve represents no current injection; the top curve corresponds to the longest possible period that the network produced with hyperpolarizing current injection before becoming quiescent; the bottom curve represents the shortest period that the network produced with depolarizing current injection before the bursts became irregular or changed to tonic spiking. Gray area shows the burst period range of the isolated AB neuron. White insets show 2 s long voltage traces of the coupled neurons for a coupling conductance of 0.1 μS and the largest hyperpolarizing current injection. Voltage in the insets is from −60 to −30 mV. A: an AB–AB model network cannot oscillate with periods longer than those of an isolated AB model neuron. For small coupling conductances (−0.3 μS) its period range was smaller than that of the isolated AB neuron. Inset: 2 phase-locked AB neurons when hyperpolarizing current (I = −0.41 nA) was applied to one of them (lighter trace). B: when current is injected into the model AB neuron, for small to moderate coupling conductance the AB–PD network can produce slower (but not faster) oscillations than the isolated AB neuron. Inset: phase-locked bursting with the PD neuron bursting slightly earlier, as hyperpolarizing current (I = −0.23 nA) was injected into the AB neuron. C: when current is injected into the model PD neuron, the AB–PD network period range of oscillations is smaller than that of the isolated AB. Inset: phase-locked bursting, with the AB neuron bursting slightly earlier, as hyperpolarizing current (I = −0.27nA) was injected into the PD neuron. Most hyperpolarized voltages in the insets are (in mV): −56.5 (A), −48.2 (B), and −52.2 (C).

FIG. 8

Currents that participate in the approximately 1-Hz oscillations in the isolated S/N compartments, and their behavior in the isolated model neuron. For each model neuron, the leftmost column shows the currents for its isolated S/N compartment, and the rightmost column shows the same currents when the S/N and A compartments are connected. Boxes in the first 3 columns display the currents at a different scale for 600-ms duration, after the first 100 ms of activity. Boxes in the fourth column display enlargements of the full 1,100 ms of activity. Parameters are as in Table 2.

FIG. 9

Currents in the S/N compartments of the AB–PD pacemaker model network in the presence of modulatory inputs. Boxes show enlargements of 600 ms of activity.