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Biophysical basis for three distinct dynamical mechanisms of action potential initiation.

Prescott SA, De Koninck Y, Sejnowski TJ - PLoS Comput. Biol. (2008)

Bottom Line: Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties.From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents.Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

View Article: PubMed Central - PubMed

Affiliation: Computational Neurobiology Laboratory, Salk Institute, La Jolla, California, United States of America. prescott@neurobio.pitt.edu

ABSTRACT
Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

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Comparison of spikes initiated through different dynamical                            mechanisms.(A) Spikes initiated through a QSC or SNIC bifurcation exhibit different                            spike amplitude variability. Data are from 2D models stimulated with                            noisy Istim                            (σnoise = 10                                µA/cm2). V-clines are                            shown for rest (red) and for one stimulus intensity (blue) although                                Istim varies continuously during                            stimulation. Spikes initiated through a QSC exhibit variable amplitudes                            (yellow shading) because variations in Istim                            affect the V-w trajectory: trajectories starting close                            to the quasi-separatrix (produced by Istim                            fluctuations just exceeding rheobase) produce smaller spikes than                            trajectories starting further from the quasi-separatrix (produced by                            larger Istim fluctuations). Spikes initiated                            through an SNIC bifurcation exhibit little variability (pink shading)                            because all trajectories follow the invariant circle once the                            heteroclinic trajectories (green curves) fuse at the moment of the SNIC                            bifurcation to form a single homoclinic orbit. Histogram shows                            distribution of voltage maxima; maxima above cutoff (*) are                            considered spikes. Distributions differed significantly between cell                            classes after normalizing by maximum or by average spike amplitude                                (p<0.005 and                            p<0.001, respectively; Kolmogorov-Smirnov test).                            (B) As predicted, class 3 (single-spiking) neurons showed significantly                            greater variability in spike amplitude than class 1 (tonic-spiking)                            neurons (p<0.001 regardless of normalization by                            peak or average; Kolmogorov-Smirnov test).                            σnoise = 10 pA.
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pcbi-1000198-g003: Comparison of spikes initiated through different dynamical mechanisms.(A) Spikes initiated through a QSC or SNIC bifurcation exhibit different spike amplitude variability. Data are from 2D models stimulated with noisy Istim (σnoise = 10 µA/cm2). V-clines are shown for rest (red) and for one stimulus intensity (blue) although Istim varies continuously during stimulation. Spikes initiated through a QSC exhibit variable amplitudes (yellow shading) because variations in Istim affect the V-w trajectory: trajectories starting close to the quasi-separatrix (produced by Istim fluctuations just exceeding rheobase) produce smaller spikes than trajectories starting further from the quasi-separatrix (produced by larger Istim fluctuations). Spikes initiated through an SNIC bifurcation exhibit little variability (pink shading) because all trajectories follow the invariant circle once the heteroclinic trajectories (green curves) fuse at the moment of the SNIC bifurcation to form a single homoclinic orbit. Histogram shows distribution of voltage maxima; maxima above cutoff (*) are considered spikes. Distributions differed significantly between cell classes after normalizing by maximum or by average spike amplitude (p<0.005 and p<0.001, respectively; Kolmogorov-Smirnov test). (B) As predicted, class 3 (single-spiking) neurons showed significantly greater variability in spike amplitude than class 1 (tonic-spiking) neurons (p<0.001 regardless of normalization by peak or average; Kolmogorov-Smirnov test). σnoise = 10 pA.

Mentions: Since model parameters were chosen to produce one or another spiking pattern, simply reproducing a given pattern is not necessarily informative—this constitutes an inverse problem akin to circular reasoning. To ensure that the model does not simply mimic spiking pattern, it must predict behaviors separate from those used to choose parameters. The model makes such a prediction: spikes initiated through different dynamical mechanisms are predicted to exhibit different variability in their amplitudes. Specifically, spikes initiated through an SNIC bifurcation should have uniform amplitudes because all suprathreshold trajectories follow the invariant circle formed when the stable manifolds (green curves on Figure 3A) fuse at the moment of the bifurcation. In contrast, spikes initiated through a QSC are predicted to have variable amplitudes that are sensitive to stimulus intensity because each trajectory (including how far it extends on the abscissa) depends on where that trajectory starts relative to the quasi-separatrix.


Biophysical basis for three distinct dynamical mechanisms of action potential initiation.

Prescott SA, De Koninck Y, Sejnowski TJ - PLoS Comput. Biol. (2008)

Comparison of spikes initiated through different dynamical                            mechanisms.(A) Spikes initiated through a QSC or SNIC bifurcation exhibit different                            spike amplitude variability. Data are from 2D models stimulated with                            noisy Istim                            (σnoise = 10                                µA/cm2). V-clines are                            shown for rest (red) and for one stimulus intensity (blue) although                                Istim varies continuously during                            stimulation. Spikes initiated through a QSC exhibit variable amplitudes                            (yellow shading) because variations in Istim                            affect the V-w trajectory: trajectories starting close                            to the quasi-separatrix (produced by Istim                            fluctuations just exceeding rheobase) produce smaller spikes than                            trajectories starting further from the quasi-separatrix (produced by                            larger Istim fluctuations). Spikes initiated                            through an SNIC bifurcation exhibit little variability (pink shading)                            because all trajectories follow the invariant circle once the                            heteroclinic trajectories (green curves) fuse at the moment of the SNIC                            bifurcation to form a single homoclinic orbit. Histogram shows                            distribution of voltage maxima; maxima above cutoff (*) are                            considered spikes. Distributions differed significantly between cell                            classes after normalizing by maximum or by average spike amplitude                                (p<0.005 and                            p<0.001, respectively; Kolmogorov-Smirnov test).                            (B) As predicted, class 3 (single-spiking) neurons showed significantly                            greater variability in spike amplitude than class 1 (tonic-spiking)                            neurons (p<0.001 regardless of normalization by                            peak or average; Kolmogorov-Smirnov test).                            σnoise = 10 pA.
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pcbi-1000198-g003: Comparison of spikes initiated through different dynamical mechanisms.(A) Spikes initiated through a QSC or SNIC bifurcation exhibit different spike amplitude variability. Data are from 2D models stimulated with noisy Istim (σnoise = 10 µA/cm2). V-clines are shown for rest (red) and for one stimulus intensity (blue) although Istim varies continuously during stimulation. Spikes initiated through a QSC exhibit variable amplitudes (yellow shading) because variations in Istim affect the V-w trajectory: trajectories starting close to the quasi-separatrix (produced by Istim fluctuations just exceeding rheobase) produce smaller spikes than trajectories starting further from the quasi-separatrix (produced by larger Istim fluctuations). Spikes initiated through an SNIC bifurcation exhibit little variability (pink shading) because all trajectories follow the invariant circle once the heteroclinic trajectories (green curves) fuse at the moment of the SNIC bifurcation to form a single homoclinic orbit. Histogram shows distribution of voltage maxima; maxima above cutoff (*) are considered spikes. Distributions differed significantly between cell classes after normalizing by maximum or by average spike amplitude (p<0.005 and p<0.001, respectively; Kolmogorov-Smirnov test). (B) As predicted, class 3 (single-spiking) neurons showed significantly greater variability in spike amplitude than class 1 (tonic-spiking) neurons (p<0.001 regardless of normalization by peak or average; Kolmogorov-Smirnov test). σnoise = 10 pA.
Mentions: Since model parameters were chosen to produce one or another spiking pattern, simply reproducing a given pattern is not necessarily informative—this constitutes an inverse problem akin to circular reasoning. To ensure that the model does not simply mimic spiking pattern, it must predict behaviors separate from those used to choose parameters. The model makes such a prediction: spikes initiated through different dynamical mechanisms are predicted to exhibit different variability in their amplitudes. Specifically, spikes initiated through an SNIC bifurcation should have uniform amplitudes because all suprathreshold trajectories follow the invariant circle formed when the stable manifolds (green curves on Figure 3A) fuse at the moment of the bifurcation. In contrast, spikes initiated through a QSC are predicted to have variable amplitudes that are sensitive to stimulus intensity because each trajectory (including how far it extends on the abscissa) depends on where that trajectory starts relative to the quasi-separatrix.

Bottom Line: Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties.From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents.Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

View Article: PubMed Central - PubMed

Affiliation: Computational Neurobiology Laboratory, Salk Institute, La Jolla, California, United States of America. prescott@neurobio.pitt.edu

ABSTRACT
Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

Show MeSH
Related in: MedlinePlus