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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 dynamicalmechanism of spike initiation.(A) Phase planes show the fast activation variable Vplotted against the slower recovery variable w.Nullclines represent all points in phase space where Vor w remain constant. V-clines(colored) were calculated at rest (red) and at the onset of stimulation(blue) (Istim is indicated beside eachcurve); w-clines do not change upon stimulation andare plotted only once (gray). Black curves show response of model withdirection of trajectory indicated by arrows. Class 1 neuron: Red andgray clines intersect at three points (red arrowheads) representingstable (s) or unstable (u) fixedpoints. Stimulation shifts that V-cline upward anddestroys two of those points, thereby allowing the system to enter alimit cycle and spike repetitively. The trajectory slows as it passesthrough constriction between blue and gray clines (yellow shading)thereby allowing the neuron to spike slowly, hence the continuousf–I curve. Class 2 neuron: Red and graycurves intersect at a single, stable fixed point. Spiking begins whenstimulation destabilizes (rather than destroys) that point. Thef–I curve is discontinuous because slowspiking is not possible without the constriction (compare with class 1neuron). Class 3 neuron: Stimulation displaces but does not destroy ordestabilize the fixed point. System variablesV,w can follow different paths tothe newly positioned fixed point: a single spike is initiated whenstimulation instantaneously displaces the quasi-separatrix (dottedcurves) 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 aroundthe head of the quasi-separatrix (*) to get to the new fixedpoint – we refer to this mechanism of spike initiation as aquasi-separatrix-crossing or QSC. Dashed black curve shows alternative,subthreshold path that would be followed if trajectory remained abovethe (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage atfixed point and at max/min of limit cycle asIstim is increased. A bifurcation representsthe transition from quiescence to repetitive spiking. Type ofbifurcation is indicated on each plot. The range ofIstim over which a QSC occurs isindicated in gray and was determined by separate simulations since a QSCis not revealed by bifurcation analysis.
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pcbi-1000198-g002: Each class of excitability is derived from a distinct dynamicalmechanism of spike initiation.(A) Phase planes show the fast activation variable Vplotted against the slower recovery variable w.Nullclines represent all points in phase space where Vor w remain constant. V-clines(colored) were calculated at rest (red) and at the onset of stimulation(blue) (Istim is indicated beside eachcurve); w-clines do not change upon stimulation andare plotted only once (gray). Black curves show response of model withdirection of trajectory indicated by arrows. Class 1 neuron: Red andgray clines intersect at three points (red arrowheads) representingstable (s) or unstable (u) fixedpoints. Stimulation shifts that V-cline upward anddestroys two of those points, thereby allowing the system to enter alimit cycle and spike repetitively. The trajectory slows as it passesthrough constriction between blue and gray clines (yellow shading)thereby allowing the neuron to spike slowly, hence the continuousf–I curve. Class 2 neuron: Red and graycurves intersect at a single, stable fixed point. Spiking begins whenstimulation destabilizes (rather than destroys) that point. Thef–I curve is discontinuous because slowspiking is not possible without the constriction (compare with class 1neuron). Class 3 neuron: Stimulation displaces but does not destroy ordestabilize the fixed point. System variablesV,w can follow different paths tothe newly positioned fixed point: a single spike is initiated whenstimulation instantaneously displaces the quasi-separatrix (dottedcurves) 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 aroundthe head of the quasi-separatrix (*) to get to the new fixedpoint – we refer to this mechanism of spike initiation as aquasi-separatrix-crossing or QSC. Dashed black curve shows alternative,subthreshold path that would be followed if trajectory remained abovethe (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage atfixed point and at max/min of limit cycle asIstim is increased. A bifurcation representsthe transition from quiescence to repetitive spiking. Type ofbifurcation is indicated on each plot. The range ofIstim over which a QSC occurs isindicated in gray and was determined by separate simulations since a QSCis not revealed by bifurcation analysis.

Mentions: Right panels of Figure 2Aillustrate the spike initiating dynamics associated with class 1 excitability.Before stimulation, the V- and w-clinesintersect at three points; the leftmost intersection constitutes a stable fixedpoint that controls membrane potential. Stimulation shifts theV-cline upwards. If the V-cline wasshifted far enough (i.e., if Istim exceededrheobase), two of the intersection points were destroyed and the class 1 modelbegan to spike repetitively. Disappearance of the two fixed points and thequalitative change in behavior that results (i.e., the transition fromquiescence to repetitive spiking) is referred to most precisely as a saddle-nodeon 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 dynamicalmechanism of spike initiation.(A) Phase planes show the fast activation variable Vplotted against the slower recovery variable w.Nullclines represent all points in phase space where Vor w remain constant. V-clines(colored) were calculated at rest (red) and at the onset of stimulation(blue) (Istim is indicated beside eachcurve); w-clines do not change upon stimulation andare plotted only once (gray). Black curves show response of model withdirection of trajectory indicated by arrows. Class 1 neuron: Red andgray clines intersect at three points (red arrowheads) representingstable (s) or unstable (u) fixedpoints. Stimulation shifts that V-cline upward anddestroys two of those points, thereby allowing the system to enter alimit cycle and spike repetitively. The trajectory slows as it passesthrough constriction between blue and gray clines (yellow shading)thereby allowing the neuron to spike slowly, hence the continuousf–I curve. Class 2 neuron: Red and graycurves intersect at a single, stable fixed point. Spiking begins whenstimulation destabilizes (rather than destroys) that point. Thef–I curve is discontinuous because slowspiking is not possible without the constriction (compare with class 1neuron). Class 3 neuron: Stimulation displaces but does not destroy ordestabilize the fixed point. System variablesV,w can follow different paths tothe newly positioned fixed point: a single spike is initiated whenstimulation instantaneously displaces the quasi-separatrix (dottedcurves) 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 aroundthe head of the quasi-separatrix (*) to get to the new fixedpoint – we refer to this mechanism of spike initiation as aquasi-separatrix-crossing or QSC. Dashed black curve shows alternative,subthreshold path that would be followed if trajectory remained abovethe (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage atfixed point and at max/min of limit cycle asIstim is increased. A bifurcation representsthe transition from quiescence to repetitive spiking. Type ofbifurcation is indicated on each plot. The range ofIstim over which a QSC occurs isindicated in gray and was determined by separate simulations since a QSCis not revealed by bifurcation analysis.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1000198-g002: Each class of excitability is derived from a distinct dynamicalmechanism of spike initiation.(A) Phase planes show the fast activation variable Vplotted against the slower recovery variable w.Nullclines represent all points in phase space where Vor w remain constant. V-clines(colored) were calculated at rest (red) and at the onset of stimulation(blue) (Istim is indicated beside eachcurve); w-clines do not change upon stimulation andare plotted only once (gray). Black curves show response of model withdirection of trajectory indicated by arrows. Class 1 neuron: Red andgray clines intersect at three points (red arrowheads) representingstable (s) or unstable (u) fixedpoints. Stimulation shifts that V-cline upward anddestroys two of those points, thereby allowing the system to enter alimit cycle and spike repetitively. The trajectory slows as it passesthrough constriction between blue and gray clines (yellow shading)thereby allowing the neuron to spike slowly, hence the continuousf–I curve. Class 2 neuron: Red and graycurves intersect at a single, stable fixed point. Spiking begins whenstimulation destabilizes (rather than destroys) that point. Thef–I curve is discontinuous because slowspiking is not possible without the constriction (compare with class 1neuron). Class 3 neuron: Stimulation displaces but does not destroy ordestabilize the fixed point. System variablesV,w can follow different paths tothe newly positioned fixed point: a single spike is initiated whenstimulation instantaneously displaces the quasi-separatrix (dottedcurves) 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 aroundthe head of the quasi-separatrix (*) to get to the new fixedpoint – we refer to this mechanism of spike initiation as aquasi-separatrix-crossing or QSC. Dashed black curve shows alternative,subthreshold path that would be followed if trajectory remained abovethe (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage atfixed point and at max/min of limit cycle asIstim is increased. A bifurcation representsthe transition from quiescence to repetitive spiking. Type ofbifurcation is indicated on each plot. The range ofIstim over which a QSC occurs isindicated in gray and was determined by separate simulations since a QSCis not revealed by bifurcation analysis.
Mentions: Right panels of Figure 2Aillustrate the spike initiating dynamics associated with class 1 excitability.Before stimulation, the V- and w-clinesintersect at three points; the leftmost intersection constitutes a stable fixedpoint that controls membrane potential. Stimulation shifts theV-cline upwards. If the V-cline wasshifted far enough (i.e., if Istim exceededrheobase), two of the intersection points were destroyed and the class 1 modelbegan to spike repetitively. Disappearance of the two fixed points and thequalitative change in behavior that results (i.e., the transition fromquiescence to repetitive spiking) is referred to most precisely as a saddle-nodeon 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