A computational investigation of electrotonic coupling between pyramidal cells in the cortex

Abstract

The existence of electrical communication among pyramidal cells (PCs) in the adult cortex has been debated by neuroscientists for several decades. Gap junctions (GJs) among cortical interneurons have been well documented experimentally and their functional roles have been proposed by both computational neuroscientists and experimentalists alike. Experimental evidence for similar junctions among pyramidal cells in the cortex, however, has remained elusive due to the apparent rarity of these couplings among neurons. In this work, we develop a neuronal network model that includes observed probabilities and strengths of electrotonic coupling between PCs and gap-junction coupling among interneurons, in addition to realistic synaptic connectivity among both populations. We use this network model to investigate the effect of electrotonic coupling between PCs on network behavior with the goal of theoretically addressing this controversy of existence and purpose of electrotonically coupled PCs in the cortex.

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Acknowledgments

We thank Jeffrey Banks, Jordan Angel, Songting Li, and Wei Dai for helpful discussions. We dedicate this paper to our late coauthor and mentor D.C.

Funding

This work was supported in part by National Key R&D Program of China (2019YFA0709503), NSFC-11671259, NSFC-11722107, SJTU-UM Collaborative Research Program, and the Student Innovation Center at Shanghai Jiao Tong University (D.Z.). This work was also supported in part by the NSF Research Training Groups (RTG) under Grant No. DMS-1344962 (G.K., J.C.); US Department of Education Graduate Assistance in Areas of National Need (GAANN), and the NSF Mathematical Sciences PostDoctoral Research Fellowship (MSPRF) under Grant No. DMS-1703761 (J.C.).

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Appendix

Appendix

In this Appendix, we describe the process by which parameter values for the FS cells and PCs in the cortical network are chosen. Included in each section is evidence that those parameter choices lead to neuron behavior similar to experimentally measured behavior for electrically coupled neurons of each type.

Table 4 Parameter ranges extracted from experiments (Galarreta and Hestrin 1999; Gibson et al. 1999; Wang et al. 2010; Pospischil et al. 2008). The ranges for the junctional conductance, gC, were determined from reported values of the conductance in units of nS and converted to mS/cm2 using a diameter of 56.9 μm for FS cells and 96 μm for PCs, as estimated in Pospischil et al. (2008)

Parameter choices

We choose parameter values for the HH model of the downstream cortical network to qualitatively match experimentally observed voltage traces of FS cells and PCs. In particular, we fix the following parameters: the capacitance C, resting voltage vR, sodium reversal potential vNa, and potassium reversal potential vK at the values shown in Table 1 in the main text, and vary all other neuron parameters within the ranges described in Table 4. Then, we choose values for these parameters within those ranges based on qualitatively matching voltage traces to experiments. Note that, due to sensitivity in measuring equipment and procedural differences in collecting data across labs, we do not attempt a more quantitative match. Rather, we demonstrate that our model neurons behave similarly to experiments in response to different external input and reproduce phenomena unique to GJ-coupled FS cells and EJ-coupled PCs in the adult cortex.

Matching experimental results: FS cells

For GJ-coupled FS cortical cells, we use the experimental results from Galarreta and Hestrin (1999) and Galarreta and Hestrin (2001) to determine the leakage conductance, gL, which controls the sub-threshold voltage decay back to rest after a depolarization or hyperpolarization; the maximal sodium and potassium conductances, \(\bar {g}_{Na}\) and \(\bar {g}_{K}\), respectively, which control the rise and fall of the action potential, respectively; and the GJ conductance, gC, which controls the amount of information that is transmitted through the junction.

We begin by investigating how the leakage conductance affects the sub-threshold voltage decay of an FS neuron after a sub-threshold depolarization and hyperpolarization. Note that since the potassium and sodium channels are essentially closed unless the neuron is firing an action potential, we set \(\bar {g}_{Na}\) and \(\bar {g}_{K}\) to zero to investigate sub-threshold effects. The experiments by Galarreta and Hestrin (1999) and Galarreta and Hestrin (2001) show that a GJ-coupled FS cell exhibits a decay time of about 50 ms in response to a constant current input for 100 ms that raises the voltage about ± 10 mV from rest for depolarization and hyperpolarization protocols. To replicate this experiment in our model, we inject a constant current to our FS cells of strength Iconst = ± 0.5 mA/cm2. Figure 10a shows the voltage traces of the FS cell in response to hyperpolarization and depolarization for several values of gL, together with a table of decay times for each value.

Fig. 10
figure10

Sub-threshold voltage traces for FS cells. a Voltage traces, together with a table of the decay time to rest, for various values of gL. The decay time is calculated as the time it takes for the voltage to decay from its value at the time the stimulus is turned off (100 ms) to within 0.5 mV of the resting potential. b Voltage traces, together with a table of values of the coupling coefficient (ratio of the voltage change in the post-junctional cell to the voltage change in the pre-junctional cell), for various values of gC. The external drive for these figures is I = ± 0.5 mA

We fix the leakage conductance at gL = 0.1 mS/cm2 and, while still considering the neuron’s sub-threshold behavior, begin to narrow down the choice for the GJ conductance, gC. Experiments show that the coupling coefficient (CC), or the ratio of the change in post-junctional voltage to the change in pre-junctional voltage, for GJ-coupled FS cells is about 10% (Galarreta and Hestrin 1999). Figure 10b shows the pre-junctional (solid curves) and post-junctional (dashed curves) voltage changes for several values of the GJ conductance, with the resulting coupling coefficient shown in the corresponding table. Note that several values of the GJ conductance (gC = 0.01—0.015) yield CC values near 10%. Therefore, we use the spiking behavior of the cells to further narrow down this parameter choice.

To match the spiking behavior of FS cells, we use voltage traces measured in Wang et al. (2010). In the model, three parameters control the action potential shape: the spiking threshold, vT; the maximal sodium conductance, \(\bar {g}_{Na}\); and the maximal potassium conductance, \(\bar {g}_{K}\). We begin with the spiking threshold, vT, keeping the sodium and potassium maximal conductances near their respective averages of 45 mS/cm2 and 5 mS/cm2.

In the experiments, the sub-threshold voltage begins a sharp increase around − 45mV. We mark this location in our model simulation and look for the value of vT that gives a sub-threshold increase until around − 45 mV and then begins its ascent to the spike; see Fig. 11a. Experimental traces show that the spike decreases to about − 62mV and increases to a height of about 20 mV, giving indicators for our choices of \(\bar {g}_{K}\) and \(\bar {g}_{Na}\), respectively. We mark those locations in our model simulations, and vary the values for the sodium and potassium conductances; see Fig. 11b and c.

Fig. 11
figure11

Action-potential model replications for FS cells. a Model action potential shape for various values of the spiking threshold, vT. The orange dotted line denotes the experimentally estimated threshold. b, c Action potential shapes as a result of various values of the maximal potassium conductance, \(\bar {g}_{K}\), and the maximal sodium conductance, \(\bar {g}_{Na}\), respectively. The orange dotted line in b and c denotes the experimentally estimated value for the maximal decay of the action potential and maximal peak of the action potential, respectively. d: Model voltage trace for pre-junctional cell action potentials in response to constant current input and the resulting spikelets in the post-junctional cell, together with a table of the spikelet amplitude, for various values of gC. The constant current input used here is I = 3.0 mS/cm2

To narrow down the choice of the GJ conductance, we consider the spikelet amplitude, or the height of the post-junctional voltage increase in response to a pre-junctional action potential; see Fig. 11d. Note that the experimentally measured spikelet amplitude is 1.5 ± 0.2 mV, as reported in Wang et al. (2010). The final parameter choices are shown in Table 1 in the main text.

Reproducing experimental phenomena: FS cells

In this section, we show that our model, with the chosen parameters, captures experimental phenomena exhibited by GJ-coupled cortical FS cells. We note that these parameter choices are not unique and the results do not vary with reasonable changes in these parameters.

The phenomena that we choose to replicate are from the experiments by Galarreta and Hestrin (1999) in which they demonstrate the behavior of two GJ-coupled FS cells by injecting current into one cell and observing the effect in the coupled cell. First, Galarreta and Hestrin show that GJs act as low-pass filters, preferentially transmitting low-frequency signals, by injecting a sine-wave current into the pre-junctional cell and showing that the CC of the post-junctional cell decreases as the frequency of the sinusoidal input increases. They also demonstrate that the spikelet in the post-junctional cell exhibits a phase lag that increases with increasing input frequency. In response to a sinusoidal input, our model captures this phenomenon as well; see Fig. 12a. In response to increasing frequency, our model behaves similarly to experiments, with an increase in phase lag, and decrease in coupling coefficient, as frequency increases; see Fig. 12b. Note that the frequency at which these curves cross occurs near the same location as in the experiments.

Fig. 12
figure12

Reproducing experimentally observed phenomena for FS cells. a Voltage responses in pre- (black) and post- (blue) junctional cells due to a sinusoidal input to the pre-junctional cell. The left axis describes the pre-junctional voltage, while the right axis shows the post-junctional voltage. b The frequency dependency of the coupling coefficient (open squares) and phase lag (closed circles) for GJ-coupled cells. c Action potentials in the pre-junctional cell (black) result in a short summation of spikelets in the post-junctional cell (blue) if the membrane potential of the post-junctional cell is hyperpolarized to a value below the resting potential (− 72mV in the model). d Superimposed action potential from the pre-junctional cell (black) with the spikelet of the post-junctional cell (blue). Axes labels are color coded for which curve they represent

Galarreta and Hestrin (1999) and Galarreta and Hestrin (2001) also demonstrated that a high-frequency train of action potentials in the pre-junctional cell is transmitted as a short summation of spikelets in the post-junctional cell when the post-junctional cell is hyperpolarized to a value below the resting potential. Our model qualitatively reproduces this phenomenon as well; see Fig. 12c. Finally, Galarreta and Hestrin demonstrate that there is a delay between the peak of the action potential in the pre-junctional cell and the peak of the spikelet in the post-junctional cell, which our model reproduces in Fig. 12d.

Matching experimental results: Pyramidal cells (PCs)

We utilize similar techniques as in the previous section for a pair of PCs coupled by an EJ with the aim of replicating experimental data described by Wang et al. (2010). We begin again with the sub-threshold dynamics by setting both \(\bar {g}_{Na}\) and \(\bar {g}_{K}\) to 0 and considering different values for the leakage conductance, gL. Experiments show that PCs have a decay time for sub-threshold voltage of about 200 ms in response to a hyperpolarization and depolarization of about ± 20 mV. We perform a similar task in our model neuron; see Fig. 13a.

Fig. 13
figure13

Sub-threshold voltage-trace model replications for PCs. a Voltage traces from model PCs, together with a table of decay times, for various values of gL. b Voltage traces from model PCs, together with a table of values for the coupling coefficient (ratio of the change in voltage of the post-junctional cell to the voltage change in the pre-junctional cell), for various values of gC. c The relationship between the amplitude of the post-junctional cell and the pre-junctional voltage amplitude for three values of the EJ conductance

Experiments show that EJ-coupled PCs have a CC of about 60%. Therefore, we vary the EJ conductance, gC, and show that there are several values of the conductance for which the coupling coefficient is near 60%; see Fig. 13b. Wang et al. (2010) also measured very little dependence of the EJ conductance on the membrane potential of either cell, which can be demonstrated by calculating the amplitude of the pre-junctional and post-junctional voltage difference from rest at several different values of the membrane potential. Our model reconstructs this independence of the junctional conductance on membrane potential as well, and we calculate the slope of the line; see Fig. 13c (0.8 as measured experimentally). As in the case of FS cells, we use the spiking behavior of PCs to narrow down the choice for the EJ conductance.

Next, we consider the spiking dynamics of a pair of EJ-coupled PCs. First, observe that action potentials have a different shape in response to a constant current input compared with a 20-Hz spike train, as shown in Wang et al. (2010). Therefore, we consider both types of input and use characteristics of the action potentials generated in each case to determine values for vT, \(\bar {g}_{Na}\), and \(\bar {g}_{K}\). Note that to simulate a spike train, we input constant-current step pulses into the pre-junctional cell at a particular frequency such that the pre-junctional cell fires one spike per current pulse. We do not include any stochasticity in these matching experiments.

The spiking threshold is chosen by considering the voltage at which the action potential begins its ascent in response to a constant-current input, as shown by the orange dotted line in Fig. 14a (top), and a high voltage of the tail of the action potential in response to a 20-Hz spiking input, as shown in Fig. 14b (bottom). The maximal sodium and potassium conductances, \(\bar {g}_{Na}\) and \(\bar {g}_{K}\), respectively, are chosen by matching spiking dynamics in the 20-Hz spike-train case with the aim of obtaining a very fast rise and a slow decay, as was observed experimentally by Wang et al. (2010).

Fig. 14
figure14

Action-potential model replications for PCs. a Model action potential shape for various values of the spiking threshold, vT, for a constant current input (top) and a 20-Hz stimulus (bottom). The orange dotted line denotes the experimentally estimated value for the voltage at which the action potential begins. b Action-potential shapes as a result of various values of the maximal potassium conductance, \(\bar {g}_{K}\), (top) and the maximal sodium conductance, \(\bar {g}_{Na}\), (bottom). c Model voltage trace of pre-junctional action potentials in response to 20-Hz input and corresponding spikelets in the post-junctional cell, together with a table of spikelet amplitudes, resulting from various values of gC

We choose the conductance of the EJ by matching the spikelet amplitude of 14 mV as measured experimentally. Notice that an EJ conductance value of 0.08 mS/cm2 gives a spikelet amplitude of 15 mV, and is within reason with respect to the sub-threshold dynamics (recall Fig. 13b and c). The final parameter set is shown in Table 1 in the main text.

Reproducing experimental phenomena: PCs

We show that our model of two EJ-coupled PCs captures experimental phenomena observed by Wang et al. (2010), including responses to high-frequency stimuli and spikelet summation. Due to the high junctional conductance, Wang et al. (2010) recorded that spikelets in the post-junctional cell sum to threshold in response to a high-frequency (70 Hz) stimulation of action potentials in the pre-junctional cell. Our model PCs mimic this behavior in response to a 70-Hz input stimulus; see Fig. 15a. To quantify this summation, Wang et al. (2010) calculated a second summation rate, the ratio of the difference in height between the second and the first spikelets to the height of the first spikelet. Wang et al. (2010) showed that, across all measured EJ-coupled PC pairs, this second-summation rate increases with an increase in stimulation frequency to the pre-junctional cell. Our model also shows this increase in summation rate; see Fig. 15b. In addition, experiments indicate that the amplitude of the post-junctional spikelet has very little dependence on the membrane potential of the post-junctional cell. Our model replicates this small dependence of the spikelet amplitude on the membrane potential of the post-junctional cell, with a slightly higher spikelet amplitude than experimentally observed in all cases; see Fig. 15c.

Fig. 15
figure15

Replicating experimental phenomena for PCs. a: Model voltage traces in response to a 70 Hz signal for pre-(black) and post- (red) junctional PCs. b: The second summation spikelet rate – a measure for the change in height from the first spikelet to the second – as a function of the stimulation frequency to the pre-junctional neuron for the model pair of PCs. c: The amplitude of the post-junctional spikelet for different voltages of the post-junctional cell. d: Action potentials in the pre-junctional cell (black, top) in response to a 20-Hz input and the response in the post-junctional cell (red, bottom) when it is depolarized to -60 mV. e: The frequency dependency of the coupling coefficient or ratio (open squares) and phase lag (closed circles) for EJ-coupled PCs. Parameters used to generate these figures are those listed in Table 1, together with the following values: for a, the constant external input is I = 6 mA and input frequency is 70 Hz; for d, the constant external drive is I = 8 mA and input frequency is 20 Hz; and for e, the constant external input is I = 10 mA

Action potentials in the pre-junctional cell result in spikelets in the post-junctional cell with an amplitude of about 14 mV: recall Fig. 14c. However, Wang et al. (2010) measured that in response to a 20-Hz spike train in the pre-junctional cell, the post-junctional cell will spike about 50% of the time that the pre-junctional cells spikes. We show that our model captures this phenomenon; see Fig. 15d for a few simulated milliseconds. If we run the simulation for 10 s and calculate the ratio of number times an action potential in the post-junctional cell occurs to the number of times an action potential in the pre-junctional cell occurs, the result is 54%. Though Wang et al. did not measure the coupling coefficient as a function of stimulation frequency, we use our model to show that EJs behave similarly to GJs in this respect; see Fig. 15e.

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Crodelle, J., Zhou, D., Kovačič, G. et al. A computational investigation of electrotonic coupling between pyramidal cells in the cortex. J Comput Neurosci 48, 387–407 (2020). https://doi.org/10.1007/s10827-020-00762-5

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Keywords

  • Neuronal networks
  • Hodgkin-Huxley model
  • Gap junctions
  • Pyramidal cells