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Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model.

Weinberg SH - PLoS ONE (2015)

Bottom Line: We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced.In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order.Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

View Article: PubMed Central - PubMed

Affiliation: Virginia Modeling, Analysis and Simulation Center, Old Dominion University, 1030 University Boulevard, Suffolk, Virginia, USA.

ABSTRACT
Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

No MeSH data available.


Related in: MedlinePlus

Spike properties in the fractional-order Hodgkin-Huxley model.(A) Bifurcation diagram of Vm, sodium current INa, and potassium current IK, showing steady-state values and limit cycle maximum and minimum, as a function of the applied current Iapp, for different values of α. (B) The critical values denoting Iapp lower and upper limits for spiking (Hopf bifurcations), I1 and I2, respectively, are indicated (fractional-order Hodgkin-Huxley model (fHH), solid lines). (C) The spike frequency (top) and amplitude (bottom) are shown as a function of Iapp and α. In B and C, values for I1, I2, and spike frequency and amplitude are shown for the first-order model with scaled conductances for comparison (dashed lines, see Fig 5 and main text for more details). In the bottom panel of C, the solid and dashed lines are nearly identical.
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pone.0126629.g006: Spike properties in the fractional-order Hodgkin-Huxley model.(A) Bifurcation diagram of Vm, sodium current INa, and potassium current IK, showing steady-state values and limit cycle maximum and minimum, as a function of the applied current Iapp, for different values of α. (B) The critical values denoting Iapp lower and upper limits for spiking (Hopf bifurcations), I1 and I2, respectively, are indicated (fractional-order Hodgkin-Huxley model (fHH), solid lines). (C) The spike frequency (top) and amplitude (bottom) are shown as a function of Iapp and α. In B and C, values for I1, I2, and spike frequency and amplitude are shown for the first-order model with scaled conductances for comparison (dashed lines, see Fig 5 and main text for more details). In the bottom panel of C, the solid and dashed lines are nearly identical.

Mentions: To further determine if the dependence of spiking properties on α is primarily a consequence of the fractional passive membrane dynamics or the subsequent modulation the ionic currents, both passive and active, we run simulations in which we assume first-order Vm dynamics but also scale the ionic current IL, INa, and IK conductances, gL, gNa, and gK, respectively, such that the peak current magnitudes are equivalent to the values for a given α in the fractional-order model, i.e., the asymptotic current peak INa and IK values in Fig 4A. Peak IL magnitude is measured as the maximum depolarizing current, since IL is both depolarizing and hyperpolarizing. For example, as α is decreased in this scaled ionic conductance first-order model, gK is decreased to account for the smaller peak IK magnitude. For Iapp = 20 μA/cm2, gNa is slightly increased, while for Iapp = 100 μA/cm2, gNa is decreased. We investigate the influence of scaling the ionic current conductances both individually and combined (Fig 5). Scaling the leak conductance gL had minimal influence on spike frequency or amplitude for all values of Iapp, due to its small amplitude relative to the other currents (Fig 5A). For small Iapp, scaling the sodium conductance gNa had minimal influence (Fig 5B). For intermediate Iapp, scaling gNa does reduce spike frequency and amplitude, however to a larger extent than observed in the fractional-order neuronal model. For a larger Iapp, scaling gNa also leads to excitation block, however with a different dependence on α as in the fractional-order model (also see Fig 6B and 6C). For a given value of Iapp, scaling gK does not alter spike amplitude and further, increases spike frequency, the opposite effect as observed in the fractional-order model (Fig 5C). When all of the conductances are scaled, the collective influence is that spike frequency increases, not decreases, as the value of α decreases (Fig 5D), demonstrating that simply scaling the magnitude of ionic currents does not reproduce the influence of fractional-order Vm dynamics. Interestingly, scaling all three conductances does fairly accurately reproduce the reduction in spike amplitude as α decreases.


Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model.

Weinberg SH - PLoS ONE (2015)

Spike properties in the fractional-order Hodgkin-Huxley model.(A) Bifurcation diagram of Vm, sodium current INa, and potassium current IK, showing steady-state values and limit cycle maximum and minimum, as a function of the applied current Iapp, for different values of α. (B) The critical values denoting Iapp lower and upper limits for spiking (Hopf bifurcations), I1 and I2, respectively, are indicated (fractional-order Hodgkin-Huxley model (fHH), solid lines). (C) The spike frequency (top) and amplitude (bottom) are shown as a function of Iapp and α. In B and C, values for I1, I2, and spike frequency and amplitude are shown for the first-order model with scaled conductances for comparison (dashed lines, see Fig 5 and main text for more details). In the bottom panel of C, the solid and dashed lines are nearly identical.
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Show All Figures
getmorefigures.php?uid=PMC4430543&req=5

pone.0126629.g006: Spike properties in the fractional-order Hodgkin-Huxley model.(A) Bifurcation diagram of Vm, sodium current INa, and potassium current IK, showing steady-state values and limit cycle maximum and minimum, as a function of the applied current Iapp, for different values of α. (B) The critical values denoting Iapp lower and upper limits for spiking (Hopf bifurcations), I1 and I2, respectively, are indicated (fractional-order Hodgkin-Huxley model (fHH), solid lines). (C) The spike frequency (top) and amplitude (bottom) are shown as a function of Iapp and α. In B and C, values for I1, I2, and spike frequency and amplitude are shown for the first-order model with scaled conductances for comparison (dashed lines, see Fig 5 and main text for more details). In the bottom panel of C, the solid and dashed lines are nearly identical.
Mentions: To further determine if the dependence of spiking properties on α is primarily a consequence of the fractional passive membrane dynamics or the subsequent modulation the ionic currents, both passive and active, we run simulations in which we assume first-order Vm dynamics but also scale the ionic current IL, INa, and IK conductances, gL, gNa, and gK, respectively, such that the peak current magnitudes are equivalent to the values for a given α in the fractional-order model, i.e., the asymptotic current peak INa and IK values in Fig 4A. Peak IL magnitude is measured as the maximum depolarizing current, since IL is both depolarizing and hyperpolarizing. For example, as α is decreased in this scaled ionic conductance first-order model, gK is decreased to account for the smaller peak IK magnitude. For Iapp = 20 μA/cm2, gNa is slightly increased, while for Iapp = 100 μA/cm2, gNa is decreased. We investigate the influence of scaling the ionic current conductances both individually and combined (Fig 5). Scaling the leak conductance gL had minimal influence on spike frequency or amplitude for all values of Iapp, due to its small amplitude relative to the other currents (Fig 5A). For small Iapp, scaling the sodium conductance gNa had minimal influence (Fig 5B). For intermediate Iapp, scaling gNa does reduce spike frequency and amplitude, however to a larger extent than observed in the fractional-order neuronal model. For a larger Iapp, scaling gNa also leads to excitation block, however with a different dependence on α as in the fractional-order model (also see Fig 6B and 6C). For a given value of Iapp, scaling gK does not alter spike amplitude and further, increases spike frequency, the opposite effect as observed in the fractional-order model (Fig 5C). When all of the conductances are scaled, the collective influence is that spike frequency increases, not decreases, as the value of α decreases (Fig 5D), demonstrating that simply scaling the magnitude of ionic currents does not reproduce the influence of fractional-order Vm dynamics. Interestingly, scaling all three conductances does fairly accurately reproduce the reduction in spike amplitude as α decreases.

Bottom Line: We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced.In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order.Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

View Article: PubMed Central - PubMed

Affiliation: Virginia Modeling, Analysis and Simulation Center, Old Dominion University, 1030 University Boulevard, Suffolk, Virginia, USA.

ABSTRACT
Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

No MeSH data available.


Related in: MedlinePlus