Figure 1.
Neuronal response characteristics of Type I and Type II neurons for Morris-Lecar and cortical pyramidal cell models.
(A) Frequency-current curve for Type I and Type II Morris-Lecar model neurons. Note that the Type I cell can fire at arbitrarily low frequencies, while the Type II cell exhibits a non-zero frequency threshold. (B,C) Frequency dependence of PRCs for Morris-Lecar model neurons with Type I and Type II response characteristics. When the PRC was computed at different neuronal firing frequencies (different curves), amplitudes of phase shifts were attenuated, and the Type II neuron showed asymmetric attenuation of the phase advance and phase delay regions. (D) Frequency-current curves for Type I (, cholinergic modulation) and Type II (
, no cholinergic modulation) cortical pyramidal model neurons. The Type I neuron could fire at arbitrarily low frequencies, while the Type II neuron exhibited a threshold frequency of approximately 8 Hz. (E) PRCs for different firing frequencies of the Type I cortical pyramidal neuron. (F) PRCs for different firing frequencies of the Type II cortical pyramidal neuron. In both models, the Type I cells exhibited global attenuation of the phase responses, while increased firing frequency evoked asymmetric attenuation in the phase delay region as compared to the phase advance region in Type II cells.
Figure 2.
Effects of modifying speed of intracellular currents upon depth of PRC delay in Type II neurons.
(A–C) Effects of modifying the speed of the potassium current in the Type II Morris-Lecar neuron, with increasing values of implying faster dynamics (
). (A) PRCs of the neuron for three sample values of
, with
. As the speed of the potassium dynamics increases, the PRC delay depths grow progressively larger. (B) Absolute value of the delay depth of the PRCs as a function of
, for four different values of
, which correspond to those in Fig. 1C. (C) Neuronal firing frequency as a function of
, for the same values of
as in panel B. Note how linear growth of
results in sub-linear growth of the frequency, indicating that the delay depth is largely determined by the speed of the potassium current relative to the spiking frequency of the neuron. (D–F) Effects of modifying the speed of the slow potassium gating variable
in the Type II cortical pyramidal cell model. (D) PRCs of the neuron for three sample values of
, with
. (E) Absolute value of the delay depth of the PRCs as a function of
, for four different values of
, which correspond to those in Fig. 1F. (F) Neuronal firing frequency as a function of
, for the same values of
as in panel E. (G–I) Effects of modifying the speed of the sodium inactivation gating variable
in the cortical pyramidal cell model. (G) PRCs of the neuron for three sample values of
, with
. (H) Absolute value of the delay depth of the PRCs as a function of
, for four different values of
, which correspond to those in Fig. 1F. (I) Neuronal firing frequency as a function of
, for the same values of
as in panel H.
Figure 3.
Type II PRC profiles with the same delay depth for different levels of external current.
(A) PRC profiles of the Type II Morris-Lecar neuron for three different values of , with
separately adjusted to induce a maximum phase delay of 0.04. (B) PRC profiles of the Type II cortical pyramidal neuron for four different values of
, with
separately adjusted to induce a maximum phase delay of 0.025. (C) PRC profiles of the Type II cortical pyramidal neuron for two different values of
, with
separately adjusted to induce a maximum phase delay of 0.025. (D) Unperturbed voltage traces as a function of oscillatory phase corresponding to the Type II Morris-Lecar PRCs in panel A. (E) Unperturbed voltage traces as a function of oscillatory phase corresponding to the Type II cortical pyramidal PRCs in panel B. (F) Unperturbed voltage traces as a function of oscillatory phase corresponding to the Type II cortical pyramidal PRCs in panel C. Note how the voltage traces are virtually identical in for the cortical pyramidal model, but not for the Morris-Lecar model. This explains why the PRCs are virtually identical for the cortical pyramidal model, but not the Morris-Lecar model.
Figure 4.
Differential effects of frequency modulation on Morris-Lecar network synchronization.
Measures of network activity for simulations of large-scale (N = 200) excitatory networks of Morris-Lecar model neurons driven with various constant applied currents (different curves) for Type I (A,C,E) and Type II (B,D,F) cells. The synaptic coupling was set to for Types I and II. (A,B) Average network firing frequency as a function of the network re-wiring parameter. (C,D) Phase-zero synchronization (as quantified by the bursting measure) versus the re-wiring parameter. (E,F) Phase locking (as measured by mean phase coherence) as a function of the re-wiring parameter.
Figure 5.
Differential effects of frequency modulation on network frequency and synchronization of cortical pyramidal cells.
(A–F) Measures of network activity for simulations of large-scale (N = 200) excitatory networks of cortical pyramidal model neurons driven with varying constant applied currents for Type I (A,B,C) and Type II (D,E,F) cells. Synaptic weight was fixed at for Type I plots and
for Type II plots. (A,D) Average network frequency as a function of the re-wiring parameter for Types I and II networks. (B,E) Phase-zero synchronization, as measured by the bursting parameter, as a function of the re-wiring parameter for Types I and II networks. (C,F) Phase locking, as measured by mean phase coherence, as a function of the re-wiring parameter for Types I and II networks. Note how Type II network synchrony tended to decrease with increasing stimulation intensity, while Type I network synchrony tended to remain the same or slightly increase with increased stimulation intensity.
Figure 6.
Differential effects of frequency modulation upon phase-zero synchronization in Types I and II cortical pyramidal cell networks.
(A,B) Phase-zero synchrony (as measured by the bursting parameter, B) of Type I and Type II cortical pyramidal neuronal networks as a function of synaptic coupling strength and the re-wiring parameter,
. The left panels show values of B for networks stimulated with a high applied current (
for Type I and
for Type II), and the middle panels show values of B for networks with a low applied current (
for Type I and
for Type II). The right panel subtracts the low-frequency values of B from the high-frequency values of B. Note the pronounced negative-difference region in the Type II plot, while the Type I plot shows almost exclusively zero or positive values of the difference. (C,D) Raster plots of the last 100 ms of simulations of high-frequency (C) and low-frequency (D) Type I networks with network parameters
and
. (E,F) Raster plots of the last 1000 ms of simulation of (E) high-frequency and (F) low-frequency Type II networks with network parameters
and
. The difference in synaptic coupling values between Type I and Type II networks was due to the fact that the Type II networks synchronized better than the Type I networks and therefore required much smaller synaptic coupling values to appreciably synchronize.
Figure 7.
Time to synchronization for differentially-driven Type II cortical pyramidal cell networks.
(A) Bursting parameter B as a function of the re-wiring parameter for four Type II networks driven with different values of applied current (). Note how for values of the re-wiring parameter greater than approximately 0.40, there was little difference among the values of B for different values of
, especially for the three largest values of
. (B) Average time taken for the bursting parameter of Type II networks with randomly-distributed initial conditions to breach 0.6. Initial conditions were randomized such that initial membrane voltage values were uniformly distributed on the interval [−70 mV, −50 mV], with gating variables set to corresponding equilibrium values. Each data point is an average of 100 simulations. Note that panel B plots the subset of values of the re-wiring parameter from panel A for which the bursting parameter assumes approximately constant values.
Figure 8.
Differential effects of frequency modulation upon phase locking in cortical pyramidal neuronal networks.
(A,B) Differences in MPC between high- and low-frequency networks as a function of network re-wiring and synaptic weight for (A) Type I and (B) Type II networks composed of cortical pyramidal cells. Values of were the same as in Fig. 6.
Figure 9.
Average network frequency was directly modulated by noise frequency in stochastic input simulations.
(A,B) Average network frequency as a function of the re-wiring parameter for various values of in (A) Type I and (B) Type II stochastic-input networks. Synaptic weight was set to
in (A) and
in (B).
Figure 10.
Differential effects of frequency modulation upon synchronization in stochastically-driven cortical pyramidal cell networks.
(A,B) Differences in bursting parameter B between high- and low-frequency networks as a function of synaptic weight and the re-wiring parameter for (A) Type I and (B) Type II networks. For Type I networks,
and
corresponded to high- and low-frequency networks, respectively, while for Type II networks,
and
corresponded to high- and low-frequency networks, respectively. (C,D) Values of the bursting parameter as a function of the re-wiring parameter for four different values of
, with synaptic coupling fixed at
in (C) and
in (D). The circled regions in plots (A) and (B) were constructed by taking the difference between the highest- and lowest-frequency data points in (C) and (D). (E,F) Differences in MPC between high- and low-frequency networks as a function of synaptic weight and the re-wiring parameter for Type I and Type II networks. (G,H) Values of the MPC for four different values of
, with synaptic coupling fixed at
in (G) and
in (H). The circled regions in plots (E) and (F) were constructed by taking the difference between the highest- and lowest-frequency data points in (G) and (H). Line colors in plots (G) and (H) correspond to the same legend as in plots (C) and (D), respectively.