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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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Common phase plane geometries associated with different parameterchanges.(A) βw controls positioning of thew-cline (i.e. voltage-dependency ofIslow). Forβw = 0mV, the clines intersect tangentially at rheobasic stimulation,which translates into an SNIC bifurcation. Forβw = −13mV, the w-cline crosses theV-cline on its middle arm, which translates into aHopf bifurcation. Forβw = −21mV, the w-cline crosses theV-cline on its left arm, meaning spike initiationis limited to a QSC. See Figure 2B for corresponding bifurcation diagrams. Thus,spike initiating dynamics (and the resulting pattern of excitability)are directly related to phase plane geometry (i.e. how the clinesintersect). (B) βm controlspositioning of the V-cline (i.e.,voltage-dependency of Ifast). Reducingβm had the same effect on phaseplane geometry as increasing βw. Thepredicted consequences for excitability are confirmed on the bifurcationdiagrams. Like Islow,Ifast may comprise more than onecurrent; therefore, differences in the voltage-dependency of the netfast current may reflect the expression of different fast currentsrather than variation in the voltage-dependency of any one current (seeFigure 4). For(B–E),βw = −10mV,γw = 13mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varyingḡfast changed the shaperather than positioning of the V-cline, but bothhad equivalent consequences for excitability. (D) Varyingḡslow also changed the shapeof the V-cline, in a slightly different manner thanḡfast, but with the sameconsequences for excitability. (E) Varyingγw, which controls the slope ofthe voltage-dependent activation curve forIslow, altered thew-cline, again, with predictable consequences forexcitability.βw = 0mV.
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pcbi-1000198-g008: Common phase plane geometries associated with different parameterchanges.(A) βw controls positioning of thew-cline (i.e. voltage-dependency ofIslow). Forβw = 0mV, the clines intersect tangentially at rheobasic stimulation,which translates into an SNIC bifurcation. Forβw = −13mV, the w-cline crosses theV-cline on its middle arm, which translates into aHopf bifurcation. Forβw = −21mV, the w-cline crosses theV-cline on its left arm, meaning spike initiationis limited to a QSC. See Figure 2B for corresponding bifurcation diagrams. Thus,spike initiating dynamics (and the resulting pattern of excitability)are directly related to phase plane geometry (i.e. how the clinesintersect). (B) βm controlspositioning of the V-cline (i.e.,voltage-dependency of Ifast). Reducingβm had the same effect on phaseplane geometry as increasing βw. Thepredicted consequences for excitability are confirmed on the bifurcationdiagrams. Like Islow,Ifast may comprise more than onecurrent; therefore, differences in the voltage-dependency of the netfast current may reflect the expression of different fast currentsrather than variation in the voltage-dependency of any one current (seeFigure 4). For(B–E),βw = −10mV,γw = 13mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varyingḡfast changed the shaperather than positioning of the V-cline, but bothhad equivalent consequences for excitability. (D) Varyingḡslow also changed the shapeof the V-cline, in a slightly different manner thanḡfast, but with the sameconsequences for excitability. (E) Varyingγw, which controls the slope ofthe voltage-dependent activation curve forIslow, altered thew-cline, again, with predictable consequences forexcitability.βw = 0mV.

Mentions: In the process of building the model (see Methods), βw was identified as animportant parameter given its capacity to convert the model between all threeclasses of excitability. The biophysical meaning ofβw is deferred until Figure 4, after its functionalsignificance has been established. See Figure 8 for the effects of changing otherparameters. Therefore, to begin, we explored the effects on the model'sf–I curve of systematically varyingβw (Figure 1D). The model exhibited class 1excitability for βw>−10 mV,but class 2 and 3 excitability coexisted for allβw<−10 mV; in otherwords, class 2 or 3 excitability was exhibited depending on stimulus intensityIstim. This is evident in Figure 1B where, in the model withβw = −13mV, rheobasic stimulation elicited a single spike while stronger stimulationelicited repetitive spiking. This pattern is characteristic of phasic-spikingspinal lamina I neurons (Figure1A) and is commonly observed in other “class 2”neurons including the squid giant axon [2], trigeminalmotoneurons [19], and fast-spiking neocortical interneurons[10],[20]. Conversely,“class 3” neurons should theoretically begin spikingrepetitively if given extremely strong stimulation. In reality, strongstimulation elicits, at most, a burst of 2–4 high frequency spikes insingle-spiking spinal lamina I neurons [14], which isconsistent with Hodgkin's classification in which class 3 neurons aresaid to “repeat only with difficulty or not atall” [5]. Responses to strong stimulation can be moreaccurately reproduced in the model by incorporating slow processes likecumulative Na+ channel inactivation, but such processes werenot included in the models analyzed here in order to keep the model as simple aspossible and because such strong stimulation is arguably unphysiological in thefirst place.


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

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

Common phase plane geometries associated with different parameterchanges.(A) βw controls positioning of thew-cline (i.e. voltage-dependency ofIslow). Forβw = 0mV, the clines intersect tangentially at rheobasic stimulation,which translates into an SNIC bifurcation. Forβw = −13mV, the w-cline crosses theV-cline on its middle arm, which translates into aHopf bifurcation. Forβw = −21mV, the w-cline crosses theV-cline on its left arm, meaning spike initiationis limited to a QSC. See Figure 2B for corresponding bifurcation diagrams. Thus,spike initiating dynamics (and the resulting pattern of excitability)are directly related to phase plane geometry (i.e. how the clinesintersect). (B) βm controlspositioning of the V-cline (i.e.,voltage-dependency of Ifast). Reducingβm had the same effect on phaseplane geometry as increasing βw. Thepredicted consequences for excitability are confirmed on the bifurcationdiagrams. Like Islow,Ifast may comprise more than onecurrent; therefore, differences in the voltage-dependency of the netfast current may reflect the expression of different fast currentsrather than variation in the voltage-dependency of any one current (seeFigure 4). For(B–E),βw = −10mV,γw = 13mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varyingḡfast changed the shaperather than positioning of the V-cline, but bothhad equivalent consequences for excitability. (D) Varyingḡslow also changed the shapeof the V-cline, in a slightly different manner thanḡfast, but with the sameconsequences for excitability. (E) Varyingγw, which controls the slope ofthe voltage-dependent activation curve forIslow, altered thew-cline, again, with predictable consequences forexcitability.βw = 0mV.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2551735&req=5

pcbi-1000198-g008: Common phase plane geometries associated with different parameterchanges.(A) βw controls positioning of thew-cline (i.e. voltage-dependency ofIslow). Forβw = 0mV, the clines intersect tangentially at rheobasic stimulation,which translates into an SNIC bifurcation. Forβw = −13mV, the w-cline crosses theV-cline on its middle arm, which translates into aHopf bifurcation. Forβw = −21mV, the w-cline crosses theV-cline on its left arm, meaning spike initiationis limited to a QSC. See Figure 2B for corresponding bifurcation diagrams. Thus,spike initiating dynamics (and the resulting pattern of excitability)are directly related to phase plane geometry (i.e. how the clinesintersect). (B) βm controlspositioning of the V-cline (i.e.,voltage-dependency of Ifast). Reducingβm had the same effect on phaseplane geometry as increasing βw. Thepredicted consequences for excitability are confirmed on the bifurcationdiagrams. Like Islow,Ifast may comprise more than onecurrent; therefore, differences in the voltage-dependency of the netfast current may reflect the expression of different fast currentsrather than variation in the voltage-dependency of any one current (seeFigure 4). For(B–E),βw = −10mV,γw = 13mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varyingḡfast changed the shaperather than positioning of the V-cline, but bothhad equivalent consequences for excitability. (D) Varyingḡslow also changed the shapeof the V-cline, in a slightly different manner thanḡfast, but with the sameconsequences for excitability. (E) Varyingγw, which controls the slope ofthe voltage-dependent activation curve forIslow, altered thew-cline, again, with predictable consequences forexcitability.βw = 0mV.
Mentions: In the process of building the model (see Methods), βw was identified as animportant parameter given its capacity to convert the model between all threeclasses of excitability. The biophysical meaning ofβw is deferred until Figure 4, after its functionalsignificance has been established. See Figure 8 for the effects of changing otherparameters. Therefore, to begin, we explored the effects on the model'sf–I curve of systematically varyingβw (Figure 1D). The model exhibited class 1excitability for βw>−10 mV,but class 2 and 3 excitability coexisted for allβw<−10 mV; in otherwords, class 2 or 3 excitability was exhibited depending on stimulus intensityIstim. This is evident in Figure 1B where, in the model withβw = −13mV, rheobasic stimulation elicited a single spike while stronger stimulationelicited repetitive spiking. This pattern is characteristic of phasic-spikingspinal lamina I neurons (Figure1A) and is commonly observed in other “class 2”neurons including the squid giant axon [2], trigeminalmotoneurons [19], and fast-spiking neocortical interneurons[10],[20]. Conversely,“class 3” neurons should theoretically begin spikingrepetitively if given extremely strong stimulation. In reality, strongstimulation elicits, at most, a burst of 2–4 high frequency spikes insingle-spiking spinal lamina I neurons [14], which isconsistent with Hodgkin's classification in which class 3 neurons aresaid to “repeat only with difficulty or not atall” [5]. Responses to strong stimulation can be moreaccurately reproduced in the model by incorporating slow processes likecumulative Na+ channel inactivation, but such processes werenot included in the models analyzed here in order to keep the model as simple aspossible and because such strong stimulation is arguably unphysiological in thefirst place.

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