Input-output relation and energy efficiency in the neuron with different spike threshold dynamics

Abstract

Neuron encodes and transmits information through generating sequences of output spikes, which is a high energy-consuming process. The spike is initiated when membrane depolarization reaches a threshold voltage. In many neurons, threshold is dynamic and depends on the rate of membrane depolarization (dV/dt) preceding a spike. Identifying the metabolic energy involved in neural coding and their relationship to threshold dynamic is critical to understanding neuronal function and evolution. Here, we use a modified Morris-Lecar model to investigate neuronal input-output property and energy efficiency associated with different spike threshold dynamics. We find that the neurons with dynamic threshold sensitive to dV/dt generate discontinuous frequency-current curve and type II phase response curve (PRC) through Hopf bifurcation, and weak noise could prohibit spiking when bifurcation just occurs. The threshold that is insensitive to dV/dt, instead, results in a continuous frequency-current curve, a type I PRC and a saddle-node on invariant circle bifurcation, and simultaneously weak noise cannot inhibit spiking. It is also shown that the bifurcation, frequency-current curve and PRC type associated with different threshold dynamics arise from the distinct subthreshold interactions of membrane currents. Further, we observe that the energy consumption of the neuron is related to its firing characteristics. The depolarization of spike threshold improves neuronal energy efficiency by reducing the overlap of Na(+) and K(+) currents during an action potential. The high energy efficiency is achieved at more depolarized spike threshold and high stimulus current. These results provide a fundamental biophysical connection that links spike threshold dynamics, input-output relation, energetics and spike initiation, which could contribute to uncover neural encoding mechanism.

Keywords: biophysical connection; energy efficiency; input-output relation; spike initiation; spike threshold dynamic.

Figure 1

f − I_in curves associated with different threshold dynamics induced by adjusting β_n. (A) The half-activation voltage β_n of activation variable n is hyperpolarized from −5 to −15mV with a step of −2mV. (B) Spike threshold as a function of dV/dt with different values of β_n. The range of dV/dt is from 0.45 to 4.5mV/ms. (C) f − I_in curves generated by the neuron with different threshold dynamics for three levels of noise. The noise amplitude is σ = 0, 0.5, and 3μA/cm², respectively.

Figure 2

Effects of weak noise on spiking trains around the bifurcation. The input current is (A) I_in = 37.3μ A/cm², (B) I_in = 37.8μ A/cm², (C) I_in = 38.72μ A/cm², (D) I_in = 40.2μ A/cm², (E) I_in = 42.2μ A/cm², and (F) I_in = 45.8μ A/cm². The values of noise amplitude σ have been indicated in each panel.

Figure 3

Mean numbers of spikes as a function of noise amplitude for each threshold dynamic. (A–F) respectively give the mean spike number N (40 trials) as noise amplitude σ is increased in the neuron for 1000 ms time interval with different values of β_n. The value of I_in indicated by blue line is below the bifurcation point I^*_in and there is no repetitive spiking generated in the neuron without noise, while the values of I_in indicated by three other colors are above the bifurcation point I^*_in.

Figure 4

PRCs of the neuron with different threshold dynamics. The neurons in (A) have an insensitive spike threshold to dV/dt, and in (B) have a sensitive threshold to dV/dt. For each threshold dynamic, we compute neuronal PRC at three different natural firing frequencies. The corresponding stimulation current is: I_in = 37.52, 39.31, and 40.87μ A/cm² for β_n = −5mV; I_in = 37.966, 39.70, and 41.33μ A/cm² for β_n = −7mV; I_in = 38.74, 40.28, and 41.94μ A/cm² for β_n = −9mV; I_in = 41.17, 41.81, and 42.87μ A/cm² for β_n = −11mV; I_in = 42.63, 43.20, and 44.25μ A/cm² for β_n = −13mV; I_in = 45.478, 45.688, and 46.5μ A/cm² for β_n = −15mV. All PRCs are computed in the case of no noise, i.e., σ = 0μ A/cm².

Figure 5

Biophysical basis of the spike initiation for different threshold dynamics. (A) shows the individual steady-state membrane currents at the subthreshold potentials. Decreasing β_n has no effects on the activations of inward I_Na and outward I_L, while it causes outward I_K to be more strongly activated by perithreshold depolarization. (B) gives the relationship between steady-state net membrane current I_SS and membrane potential V (i.e., I_SS − V curve). I_SS is computed as the sum of three individual currents, i.e., I_SS = I_Na + I_K + I_L. (C,D) summarize the bifurcation diagram associated with each spike threshold dynamic. The stable equilibrium is indicated by orange solid line and unstable is orange dotted line. The stable limit cycle is indicated by green solid line and unstable is purple dotted line.

Figure 6

Ionic currents and energy consumption involved in a spike. (A) shows an action potential generated in the neuron with β_n = −5mV. (B) gives the Na⁺, K⁺ and leak currents in this action potential. The Na⁺ current is negative but we plot it with a positive sign. (C) shows the energy consumption rate for each ionic current, and (D) gives the total energy consumption rate of the action potential. The stimulus is I_in = 37.5μ A/cm² and σ = 0μ A/cm². In this case, the neuron generates repetitive spiking at about 23.5 Hz.

Figure 7

Energy consumption as a function of I_in associated with each threshold dynamic. Left panels give the average energy consumption rate of the neuron with different spike threshold dynamics for three levels of noise. The energy consumption rate is averaged over the 7000 ms time interval. Right panels are the total electrochemical energy consumed by an action potential related to each spike threshold dynamic and input current I_in. The noise amplitude is (A) σ = 0μ A/cm², (B) σ = 0.5μ A/cm², and (C) σ = 3μ A/cm².

Figure 8

Depolarizing spike threshold increases energy efficiency by reducing overlaps between Na⁺ and K⁺ currents. (A) shows the overlap Na⁺ load for different spike threshold dynamics in the case of high stimulus. (B) gives the corresponding total energy required by a spike for each spike threshold dynamic. The stimulus current is I_in = 60μ A/cm² and I_in = 70μ A/cm², the noise amplitude is σ = 0μ A/cm².