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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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Each class of excitability is derived from a distinct dynamical                            mechanism of spike initiation.(A) Phase planes show the fast activation variable V                            plotted against the slower recovery variable w.                            Nullclines represent all points in phase space where V                            or w remain constant. V-clines                            (colored) were calculated at rest (red) and at the onset of stimulation                            (blue) (Istim is indicated beside each                            curve); w-clines do not change upon stimulation and                            are plotted only once (gray). Black curves show response of model with                            direction of trajectory indicated by arrows. Class 1 neuron: Red and                            gray clines intersect at three points (red arrowheads) representing                            stable (s) or unstable (u) fixed                            points. Stimulation shifts that V-cline upward and                            destroys two of those points, thereby allowing the system to enter a                            limit cycle and spike repetitively. The trajectory slows as it passes                            through constriction between blue and gray clines (yellow shading)                            thereby allowing the neuron to spike slowly, hence the continuous                                f–I curve. Class 2 neuron: Red and gray                            curves intersect at a single, stable fixed point. Spiking begins when                            stimulation destabilizes (rather than destroys) that point. The                                f–I curve is discontinuous because slow                            spiking is not possible without the constriction (compare with class 1                            neuron). Class 3 neuron: Stimulation displaces but does not destroy or                            destabilize the fixed point. System variables                                V,w can follow different paths to                            the newly positioned fixed point: a single spike is initiated when                            stimulation instantaneously displaces the quasi-separatrix (dotted                            curves) so that the system, which existed above the (red)                            quasi-separatrix prior to stimulation, finds itself below the (blue)                            quasi-separatrix once stimulation begins; the trajectory must go around                            the head of the quasi-separatrix (*) to get to the new fixed                            point – we refer to this mechanism of spike initiation as a                            quasi-separatrix-crossing or QSC. Dashed black curve shows alternative,                            subthreshold path that would be followed if trajectory remained above                            the (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage at                            fixed point and at max/min of limit cycle as                            Istim is increased. A bifurcation represents                            the transition from quiescence to repetitive spiking. Type of                            bifurcation is indicated on each plot. The range of                                Istim over which a QSC occurs is                            indicated in gray and was determined by separate simulations since a QSC                            is not revealed by bifurcation analysis.
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pcbi-1000198-g002: Each class of excitability is derived from a distinct dynamical mechanism of spike initiation.(A) Phase planes show the fast activation variable V plotted against the slower recovery variable w. Nullclines represent all points in phase space where V or w remain constant. V-clines (colored) were calculated at rest (red) and at the onset of stimulation (blue) (Istim is indicated beside each curve); w-clines do not change upon stimulation and are plotted only once (gray). Black curves show response of model with direction of trajectory indicated by arrows. Class 1 neuron: Red and gray clines intersect at three points (red arrowheads) representing stable (s) or unstable (u) fixed points. Stimulation shifts that V-cline upward and destroys two of those points, thereby allowing the system to enter a limit cycle and spike repetitively. The trajectory slows as it passes through constriction between blue and gray clines (yellow shading) thereby allowing the neuron to spike slowly, hence the continuous f–I curve. Class 2 neuron: Red and gray curves intersect at a single, stable fixed point. Spiking begins when stimulation destabilizes (rather than destroys) that point. The f–I curve is discontinuous because slow spiking is not possible without the constriction (compare with class 1 neuron). Class 3 neuron: Stimulation displaces but does not destroy or destabilize the fixed point. System variables V,w can follow different paths to the newly positioned fixed point: a single spike is initiated when stimulation instantaneously displaces the quasi-separatrix (dotted curves) so that the system, which existed above the (red) quasi-separatrix prior to stimulation, finds itself below the (blue) quasi-separatrix once stimulation begins; the trajectory must go around the head of the quasi-separatrix (*) to get to the new fixed point – we refer to this mechanism of spike initiation as a quasi-separatrix-crossing or QSC. Dashed black curve shows alternative, subthreshold path that would be followed if trajectory remained above the (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage at fixed point and at max/min of limit cycle as Istim is increased. A bifurcation represents the transition from quiescence to repetitive spiking. Type of bifurcation is indicated on each plot. The range of Istim over which a QSC occurs is indicated in gray and was determined by separate simulations since a QSC is not revealed by bifurcation analysis.

Mentions: Right panels of Figure 2A illustrate the spike initiating dynamics associated with class 1 excitability. Before stimulation, the V- and w-clines intersect at three points; the leftmost intersection constitutes a stable fixed point that controls membrane potential. Stimulation shifts the V-cline upwards. If the V-cline was shifted far enough (i.e., if Istim exceeded rheobase), two of the intersection points were destroyed and the class 1 model began to spike repetitively. Disappearance of the two fixed points and the qualitative change in behavior that results (i.e., the transition from quiescence to repetitive spiking) is referred to most precisely as a saddle-node on invariant circle (SNIC) bifurcation [13].


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

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

Each class of excitability is derived from a distinct dynamical                            mechanism of spike initiation.(A) Phase planes show the fast activation variable V                            plotted against the slower recovery variable w.                            Nullclines represent all points in phase space where V                            or w remain constant. V-clines                            (colored) were calculated at rest (red) and at the onset of stimulation                            (blue) (Istim is indicated beside each                            curve); w-clines do not change upon stimulation and                            are plotted only once (gray). Black curves show response of model with                            direction of trajectory indicated by arrows. Class 1 neuron: Red and                            gray clines intersect at three points (red arrowheads) representing                            stable (s) or unstable (u) fixed                            points. Stimulation shifts that V-cline upward and                            destroys two of those points, thereby allowing the system to enter a                            limit cycle and spike repetitively. The trajectory slows as it passes                            through constriction between blue and gray clines (yellow shading)                            thereby allowing the neuron to spike slowly, hence the continuous                                f–I curve. Class 2 neuron: Red and gray                            curves intersect at a single, stable fixed point. Spiking begins when                            stimulation destabilizes (rather than destroys) that point. The                                f–I curve is discontinuous because slow                            spiking is not possible without the constriction (compare with class 1                            neuron). Class 3 neuron: Stimulation displaces but does not destroy or                            destabilize the fixed point. System variables                                V,w can follow different paths to                            the newly positioned fixed point: a single spike is initiated when                            stimulation instantaneously displaces the quasi-separatrix (dotted                            curves) so that the system, which existed above the (red)                            quasi-separatrix prior to stimulation, finds itself below the (blue)                            quasi-separatrix once stimulation begins; the trajectory must go around                            the head of the quasi-separatrix (*) to get to the new fixed                            point – we refer to this mechanism of spike initiation as a                            quasi-separatrix-crossing or QSC. Dashed black curve shows alternative,                            subthreshold path that would be followed if trajectory remained above                            the (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage at                            fixed point and at max/min of limit cycle as                            Istim is increased. A bifurcation represents                            the transition from quiescence to repetitive spiking. Type of                            bifurcation is indicated on each plot. The range of                                Istim over which a QSC occurs is                            indicated in gray and was determined by separate simulations since a QSC                            is not revealed by bifurcation analysis.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2551735&req=5

pcbi-1000198-g002: Each class of excitability is derived from a distinct dynamical mechanism of spike initiation.(A) Phase planes show the fast activation variable V plotted against the slower recovery variable w. Nullclines represent all points in phase space where V or w remain constant. V-clines (colored) were calculated at rest (red) and at the onset of stimulation (blue) (Istim is indicated beside each curve); w-clines do not change upon stimulation and are plotted only once (gray). Black curves show response of model with direction of trajectory indicated by arrows. Class 1 neuron: Red and gray clines intersect at three points (red arrowheads) representing stable (s) or unstable (u) fixed points. Stimulation shifts that V-cline upward and destroys two of those points, thereby allowing the system to enter a limit cycle and spike repetitively. The trajectory slows as it passes through constriction between blue and gray clines (yellow shading) thereby allowing the neuron to spike slowly, hence the continuous f–I curve. Class 2 neuron: Red and gray curves intersect at a single, stable fixed point. Spiking begins when stimulation destabilizes (rather than destroys) that point. The f–I curve is discontinuous because slow spiking is not possible without the constriction (compare with class 1 neuron). Class 3 neuron: Stimulation displaces but does not destroy or destabilize the fixed point. System variables V,w can follow different paths to the newly positioned fixed point: a single spike is initiated when stimulation instantaneously displaces the quasi-separatrix (dotted curves) so that the system, which existed above the (red) quasi-separatrix prior to stimulation, finds itself below the (blue) quasi-separatrix once stimulation begins; the trajectory must go around the head of the quasi-separatrix (*) to get to the new fixed point – we refer to this mechanism of spike initiation as a quasi-separatrix-crossing or QSC. Dashed black curve shows alternative, subthreshold path that would be followed if trajectory remained above the (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage at fixed point and at max/min of limit cycle as Istim is increased. A bifurcation represents the transition from quiescence to repetitive spiking. Type of bifurcation is indicated on each plot. The range of Istim over which a QSC occurs is indicated in gray and was determined by separate simulations since a QSC is not revealed by bifurcation analysis.
Mentions: Right panels of Figure 2A illustrate the spike initiating dynamics associated with class 1 excitability. Before stimulation, the V- and w-clines intersect at three points; the leftmost intersection constitutes a stable fixed point that controls membrane potential. Stimulation shifts the V-cline upwards. If the V-cline was shifted far enough (i.e., if Istim exceeded rheobase), two of the intersection points were destroyed and the class 1 model began to spike repetitively. Disappearance of the two fixed points and the qualitative change in behavior that results (i.e., the transition from quiescence to repetitive spiking) is referred to most precisely as a saddle-node on invariant circle (SNIC) bifurcation [13].

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