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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 parameter                            changes.(A) βw controls positioning of the                                w-cline (i.e. voltage-dependency of                                Islow). For                                βw = 0                            mV, the clines intersect tangentially at rheobasic stimulation,                            which translates into an SNIC bifurcation. For                                βw = −13                            mV, the w-cline crosses the                            V-cline on its middle arm, which translates into a                            Hopf bifurcation. For                            βw = −21                            mV, the w-cline crosses the                            V-cline on its left arm, meaning spike initiation                            is 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 clines                            intersect). (B) βm controls                            positioning of the V-cline (i.e.,                            voltage-dependency of Ifast). Reducing                                βm had the same effect on phase                            plane geometry as increasing βw. The                            predicted consequences for excitability are confirmed on the bifurcation                            diagrams. Like Islow,                                Ifast may comprise more than one                            current; therefore, differences in the voltage-dependency of the net                            fast current may reflect the expression of different fast currents                            rather than variation in the voltage-dependency of any one current (see                                Figure 4). For                            (B–E),                            βw = −10                            mV,                            γw = 13                            mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varying                                ḡfast changed the shape                            rather than positioning of the V-cline, but both                            had equivalent consequences for excitability. (D) Varying                                ḡslow also changed the shape                            of the V-cline, in a slightly different manner than                                ḡfast, but with the same                            consequences for excitability. (E) Varying                                γw, which controls the slope of                            the voltage-dependent activation curve for                            Islow, altered the                            w-cline, again, with predictable consequences for                            excitability.                            βw = 0                            mV.
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pcbi-1000198-g008: Common phase plane geometries associated with different parameter changes.(A) βw controls positioning of the w-cline (i.e. voltage-dependency of Islow). For βw = 0 mV, the clines intersect tangentially at rheobasic stimulation, which translates into an SNIC bifurcation. For βw = −13 mV, the w-cline crosses the V-cline on its middle arm, which translates into a Hopf bifurcation. For βw = −21 mV, the w-cline crosses the V-cline on its left arm, meaning spike initiation is 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 clines intersect). (B) βm controls positioning of the V-cline (i.e., voltage-dependency of Ifast). Reducing βm had the same effect on phase plane geometry as increasing βw. The predicted consequences for excitability are confirmed on the bifurcation diagrams. Like Islow, Ifast may comprise more than one current; therefore, differences in the voltage-dependency of the net fast current may reflect the expression of different fast currents rather than variation in the voltage-dependency of any one current (see Figure 4). For (B–E), βw = −10 mV, γw = 13 mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varying ḡfast changed the shape rather than positioning of the V-cline, but both had equivalent consequences for excitability. (D) Varying ḡslow also changed the shape of the V-cline, in a slightly different manner than ḡfast, but with the same consequences for excitability. (E) Varying γw, which controls the slope of the voltage-dependent activation curve for Islow, altered the w-cline, again, with predictable consequences for excitability. βw = 0 mV.

Mentions: In the process of building the model (see Methods), βw was identified as an important parameter given its capacity to convert the model between all three classes of excitability. The biophysical meaning of βw is deferred until Figure 4, after its functional significance has been established. See Figure 8 for the effects of changing other parameters. Therefore, to begin, we explored the effects on the model's f–I curve of systematically varying βw (Figure 1D). The model exhibited class 1 excitability for βw>−10 mV, but class 2 and 3 excitability coexisted for all βw<−10 mV; in other words, class 2 or 3 excitability was exhibited depending on stimulus intensity Istim. This is evident in Figure 1B where, in the model with βw = −13 mV, rheobasic stimulation elicited a single spike while stronger stimulation elicited repetitive spiking. This pattern is characteristic of phasic-spiking spinal lamina I neurons (Figure 1A) and is commonly observed in other “class 2” neurons including the squid giant axon [2], trigeminal motoneurons [19], and fast-spiking neocortical interneurons [10],[20]. Conversely, “class 3” neurons should theoretically begin spiking repetitively if given extremely strong stimulation. In reality, strong stimulation elicits, at most, a burst of 2–4 high frequency spikes in single-spiking spinal lamina I neurons [14], which is consistent with Hodgkin's classification in which class 3 neurons are said to “repeat only with difficulty or not at all” [5]. Responses to strong stimulation can be more accurately reproduced in the model by incorporating slow processes like cumulative Na+ channel inactivation, but such processes were not included in the models analyzed here in order to keep the model as simple as possible and because such strong stimulation is arguably unphysiological in the first 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 parameter                            changes.(A) βw controls positioning of the                                w-cline (i.e. voltage-dependency of                                Islow). For                                βw = 0                            mV, the clines intersect tangentially at rheobasic stimulation,                            which translates into an SNIC bifurcation. For                                βw = −13                            mV, the w-cline crosses the                            V-cline on its middle arm, which translates into a                            Hopf bifurcation. For                            βw = −21                            mV, the w-cline crosses the                            V-cline on its left arm, meaning spike initiation                            is 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 clines                            intersect). (B) βm controls                            positioning of the V-cline (i.e.,                            voltage-dependency of Ifast). Reducing                                βm had the same effect on phase                            plane geometry as increasing βw. The                            predicted consequences for excitability are confirmed on the bifurcation                            diagrams. Like Islow,                                Ifast may comprise more than one                            current; therefore, differences in the voltage-dependency of the net                            fast current may reflect the expression of different fast currents                            rather than variation in the voltage-dependency of any one current (see                                Figure 4). For                            (B–E),                            βw = −10                            mV,                            γw = 13                            mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varying                                ḡfast changed the shape                            rather than positioning of the V-cline, but both                            had equivalent consequences for excitability. (D) Varying                                ḡslow also changed the shape                            of the V-cline, in a slightly different manner than                                ḡfast, but with the same                            consequences for excitability. (E) Varying                                γw, which controls the slope of                            the voltage-dependent activation curve for                            Islow, altered the                            w-cline, again, with predictable consequences for                            excitability.                            βw = 0                            mV.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2551735&req=5

pcbi-1000198-g008: Common phase plane geometries associated with different parameter changes.(A) βw controls positioning of the w-cline (i.e. voltage-dependency of Islow). For βw = 0 mV, the clines intersect tangentially at rheobasic stimulation, which translates into an SNIC bifurcation. For βw = −13 mV, the w-cline crosses the V-cline on its middle arm, which translates into a Hopf bifurcation. For βw = −21 mV, the w-cline crosses the V-cline on its left arm, meaning spike initiation is 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 clines intersect). (B) βm controls positioning of the V-cline (i.e., voltage-dependency of Ifast). Reducing βm had the same effect on phase plane geometry as increasing βw. The predicted consequences for excitability are confirmed on the bifurcation diagrams. Like Islow, Ifast may comprise more than one current; therefore, differences in the voltage-dependency of the net fast current may reflect the expression of different fast currents rather than variation in the voltage-dependency of any one current (see Figure 4). For (B–E), βw = −10 mV, γw = 13 mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varying ḡfast changed the shape rather than positioning of the V-cline, but both had equivalent consequences for excitability. (D) Varying ḡslow also changed the shape of the V-cline, in a slightly different manner than ḡfast, but with the same consequences for excitability. (E) Varying γw, which controls the slope of the voltage-dependent activation curve for Islow, altered the w-cline, again, with predictable consequences for excitability. βw = 0 mV.
Mentions: In the process of building the model (see Methods), βw was identified as an important parameter given its capacity to convert the model between all three classes of excitability. The biophysical meaning of βw is deferred until Figure 4, after its functional significance has been established. See Figure 8 for the effects of changing other parameters. Therefore, to begin, we explored the effects on the model's f–I curve of systematically varying βw (Figure 1D). The model exhibited class 1 excitability for βw>−10 mV, but class 2 and 3 excitability coexisted for all βw<−10 mV; in other words, class 2 or 3 excitability was exhibited depending on stimulus intensity Istim. This is evident in Figure 1B where, in the model with βw = −13 mV, rheobasic stimulation elicited a single spike while stronger stimulation elicited repetitive spiking. This pattern is characteristic of phasic-spiking spinal lamina I neurons (Figure 1A) and is commonly observed in other “class 2” neurons including the squid giant axon [2], trigeminal motoneurons [19], and fast-spiking neocortical interneurons [10],[20]. Conversely, “class 3” neurons should theoretically begin spiking repetitively if given extremely strong stimulation. In reality, strong stimulation elicits, at most, a burst of 2–4 high frequency spikes in single-spiking spinal lamina I neurons [14], which is consistent with Hodgkin's classification in which class 3 neurons are said to “repeat only with difficulty or not at all” [5]. Responses to strong stimulation can be more accurately reproduced in the model by incorporating slow processes like cumulative Na+ channel inactivation, but such processes were not included in the models analyzed here in order to keep the model as simple as possible and because such strong stimulation is arguably unphysiological in the first 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