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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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Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels showhow they combine to produce the instantaneous(Iinst) and steady state(Iss) I–Vcurves. Double-headed arrows highlight effect ofβw on the voltage-dependency ofIslow. Class 3 neuron:Islow activates at lowerV than Ifast, meaning slownegative feedback keeps V from increasing high enoughto initiate fast positive feedback at steady state.Fast positive feedback (that results in a spike) can be initiated onlyif the system is perturbed from steady state. Quasi-separatrix (blue)has a region of negative slope (*) indicating where net positivefeedback occurs given the kinetic difference between fast and slowcurrents: positive feedback that activates rapidly can competeeffectively with stronger negative feedback whose full activation isdelayed by its slower kinetics. If V is forced rapidlypast the blue arrowhead, fast positive feedback initiates a single spikebefore slow negative feedback catches up and forces the system back toits stable fixed point. Quasi-separatrix is plotted as the sum of allcurrents but with Islow calculated as afunction of w at the quasi-separatrix (see phase planein Figure 2A) ratherthan at steady state and is shown here forIstim = 60µA/cm2. Class 2 neuron:Islow andIfast activate at roughly the sameV. A Hopf bifurcation occurs at the point indicated bythe arrow, where  (see Results).This means that fast positive feedback exceeds slow negative feedback atsteady state; as for class 3 neurons, this relies on positive feedbackhaving fast kinetics since the net perithreshold current is stilloutward (i.e., steady state I–V curve ismonotonic). Note that the slope of the steady-stateI–V curve is less steep in the class 2model than in the class 3 model. Class 1 neuron:Islow activates at higherV than Ifast, meaning slownegative feedback does not begin activating until after the spike isinitiated. This gives a steady stateI–V curve that isnon-monotonic with a region of negative slope (*) near the apexof the instantaneous I–Vcurve. The SNIC bifurcation occurs when∂Iss/∂t = 0(arrowhead) because, at this voltage, Ifastcounterbalances Ileak and any furtherdepolarization will cause progressive activation ofIfast. (B) Changingḡfast in the 2D model hadequivalent effects on the shape of the steady stateI–V curves. Unlike in (A), voltage at theapex of the instantaneousI–V curve (purple arrows)changes as ḡfast is varied; inother words, the net current at perithreshold potentials can bemodulated by changing fast currents (which directly impact voltagethreshold) rather than by changing the amplitude or voltage-dependencyof slow currents. This is consistent with results in Figure 8. (C) Speedingup the kinetics of Islow impacts the onsetof class 2 and 3 excitability. Compared with original model(φw = 0.15;black), increasing φw to 0.25(red) increased Istim required to cause aHopf bifurcation or a QSC, but did not affectIstim required to cause an SNICbifurcation; reducing φw to 0.10(green) had the opposite effect (summarized in right panel). Increasingφw also widened thediscontinuity in the class 2 f–I curve andallowed class 2 and 3 neurons to achieve higher spiking rates withstrong Istim because of the faster recoverybetween spikes; reducing φw hadthe opposite effects.
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pcbi-1000198-g009: Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels showhow they combine to produce the instantaneous(Iinst) and steady state(Iss) I–Vcurves. Double-headed arrows highlight effect ofβw on the voltage-dependency ofIslow. Class 3 neuron:Islow activates at lowerV than Ifast, meaning slownegative feedback keeps V from increasing high enoughto initiate fast positive feedback at steady state.Fast positive feedback (that results in a spike) can be initiated onlyif the system is perturbed from steady state. Quasi-separatrix (blue)has a region of negative slope (*) indicating where net positivefeedback occurs given the kinetic difference between fast and slowcurrents: positive feedback that activates rapidly can competeeffectively with stronger negative feedback whose full activation isdelayed by its slower kinetics. If V is forced rapidlypast the blue arrowhead, fast positive feedback initiates a single spikebefore slow negative feedback catches up and forces the system back toits stable fixed point. Quasi-separatrix is plotted as the sum of allcurrents but with Islow calculated as afunction of w at the quasi-separatrix (see phase planein Figure 2A) ratherthan at steady state and is shown here forIstim = 60µA/cm2. Class 2 neuron:Islow andIfast activate at roughly the sameV. A Hopf bifurcation occurs at the point indicated bythe arrow, where (see Results).This means that fast positive feedback exceeds slow negative feedback atsteady state; as for class 3 neurons, this relies on positive feedbackhaving fast kinetics since the net perithreshold current is stilloutward (i.e., steady state I–V curve ismonotonic). Note that the slope of the steady-stateI–V curve is less steep in the class 2model than in the class 3 model. Class 1 neuron:Islow activates at higherV than Ifast, meaning slownegative feedback does not begin activating until after the spike isinitiated. This gives a steady stateI–V curve that isnon-monotonic with a region of negative slope (*) near the apexof the instantaneous I–Vcurve. The SNIC bifurcation occurs when∂Iss/∂t = 0(arrowhead) because, at this voltage, Ifastcounterbalances Ileak and any furtherdepolarization will cause progressive activation ofIfast. (B) Changingḡfast in the 2D model hadequivalent effects on the shape of the steady stateI–V curves. Unlike in (A), voltage at theapex of the instantaneousI–V curve (purple arrows)changes as ḡfast is varied; inother words, the net current at perithreshold potentials can bemodulated by changing fast currents (which directly impact voltagethreshold) rather than by changing the amplitude or voltage-dependencyof slow currents. This is consistent with results in Figure 8. (C) Speedingup the kinetics of Islow impacts the onsetof class 2 and 3 excitability. Compared with original model(φw = 0.15;black), increasing φw to 0.25(red) increased Istim required to cause aHopf bifurcation or a QSC, but did not affectIstim required to cause an SNICbifurcation; reducing φw to 0.10(green) had the opposite effect (summarized in right panel). Increasingφw also widened thediscontinuity in the class 2 f–I curve andallowed class 2 and 3 neurons to achieve higher spiking rates withstrong Istim because of the faster recoverybetween spikes; reducing φw hadthe opposite effects.

Mentions: Interpretation of the phase plane geometry can be formalized by doing localstability analysis near the fixed points ([27], see also chapter11 in [28]). In class 3 neurons, at the stable fixed point. This means, at steadystate, that positive feedback is slower than the rate of negativefeedback,φw/τw.Subthreshold activation of Islow produces a steadystate I–V curve that is monotonic and sufficientlysteep near the apex of the instantaneous I–V curvethat V is prohibited from rising high enough to stronglyactivate Ifast (Figure 9A, left). However, because the twofeedback processes have different kinetics, a spike can be initiated if thesystem is perturbed from steady state: if V escapes high enoughto activate Ifast (e.g., at the onset of an abruptstep in Istim), fast-activating inward current canoverpower slow-activating outward current—the latter is stronger whenfully activated, but can only partially activate (because of its slow kinetics)before a spike is inevitable. Through this mechanism, a single spike can beinitiated before negative feedback forces the system back to its stable fixedpoint, hence class 3 excitability. Speeding up the kinetics ofIslow predictably allowsIslow to compete more effectively withIfast (see below).


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

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

Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels showhow they combine to produce the instantaneous(Iinst) and steady state(Iss) I–Vcurves. Double-headed arrows highlight effect ofβw on the voltage-dependency ofIslow. Class 3 neuron:Islow activates at lowerV than Ifast, meaning slownegative feedback keeps V from increasing high enoughto initiate fast positive feedback at steady state.Fast positive feedback (that results in a spike) can be initiated onlyif the system is perturbed from steady state. Quasi-separatrix (blue)has a region of negative slope (*) indicating where net positivefeedback occurs given the kinetic difference between fast and slowcurrents: positive feedback that activates rapidly can competeeffectively with stronger negative feedback whose full activation isdelayed by its slower kinetics. If V is forced rapidlypast the blue arrowhead, fast positive feedback initiates a single spikebefore slow negative feedback catches up and forces the system back toits stable fixed point. Quasi-separatrix is plotted as the sum of allcurrents but with Islow calculated as afunction of w at the quasi-separatrix (see phase planein Figure 2A) ratherthan at steady state and is shown here forIstim = 60µA/cm2. Class 2 neuron:Islow andIfast activate at roughly the sameV. A Hopf bifurcation occurs at the point indicated bythe arrow, where  (see Results).This means that fast positive feedback exceeds slow negative feedback atsteady state; as for class 3 neurons, this relies on positive feedbackhaving fast kinetics since the net perithreshold current is stilloutward (i.e., steady state I–V curve ismonotonic). Note that the slope of the steady-stateI–V curve is less steep in the class 2model than in the class 3 model. Class 1 neuron:Islow activates at higherV than Ifast, meaning slownegative feedback does not begin activating until after the spike isinitiated. This gives a steady stateI–V curve that isnon-monotonic with a region of negative slope (*) near the apexof the instantaneous I–Vcurve. The SNIC bifurcation occurs when∂Iss/∂t = 0(arrowhead) because, at this voltage, Ifastcounterbalances Ileak and any furtherdepolarization will cause progressive activation ofIfast. (B) Changingḡfast in the 2D model hadequivalent effects on the shape of the steady stateI–V curves. Unlike in (A), voltage at theapex of the instantaneousI–V curve (purple arrows)changes as ḡfast is varied; inother words, the net current at perithreshold potentials can bemodulated by changing fast currents (which directly impact voltagethreshold) rather than by changing the amplitude or voltage-dependencyof slow currents. This is consistent with results in Figure 8. (C) Speedingup the kinetics of Islow impacts the onsetof class 2 and 3 excitability. Compared with original model(φw = 0.15;black), increasing φw to 0.25(red) increased Istim required to cause aHopf bifurcation or a QSC, but did not affectIstim required to cause an SNICbifurcation; reducing φw to 0.10(green) had the opposite effect (summarized in right panel). Increasingφw also widened thediscontinuity in the class 2 f–I curve andallowed class 2 and 3 neurons to achieve higher spiking rates withstrong Istim because of the faster recoverybetween spikes; reducing φw hadthe opposite effects.
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Related In: Results  -  Collection

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pcbi-1000198-g009: Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels showhow they combine to produce the instantaneous(Iinst) and steady state(Iss) I–Vcurves. Double-headed arrows highlight effect ofβw on the voltage-dependency ofIslow. Class 3 neuron:Islow activates at lowerV than Ifast, meaning slownegative feedback keeps V from increasing high enoughto initiate fast positive feedback at steady state.Fast positive feedback (that results in a spike) can be initiated onlyif the system is perturbed from steady state. Quasi-separatrix (blue)has a region of negative slope (*) indicating where net positivefeedback occurs given the kinetic difference between fast and slowcurrents: positive feedback that activates rapidly can competeeffectively with stronger negative feedback whose full activation isdelayed by its slower kinetics. If V is forced rapidlypast the blue arrowhead, fast positive feedback initiates a single spikebefore slow negative feedback catches up and forces the system back toits stable fixed point. Quasi-separatrix is plotted as the sum of allcurrents but with Islow calculated as afunction of w at the quasi-separatrix (see phase planein Figure 2A) ratherthan at steady state and is shown here forIstim = 60µA/cm2. Class 2 neuron:Islow andIfast activate at roughly the sameV. A Hopf bifurcation occurs at the point indicated bythe arrow, where (see Results).This means that fast positive feedback exceeds slow negative feedback atsteady state; as for class 3 neurons, this relies on positive feedbackhaving fast kinetics since the net perithreshold current is stilloutward (i.e., steady state I–V curve ismonotonic). Note that the slope of the steady-stateI–V curve is less steep in the class 2model than in the class 3 model. Class 1 neuron:Islow activates at higherV than Ifast, meaning slownegative feedback does not begin activating until after the spike isinitiated. This gives a steady stateI–V curve that isnon-monotonic with a region of negative slope (*) near the apexof the instantaneous I–Vcurve. The SNIC bifurcation occurs when∂Iss/∂t = 0(arrowhead) because, at this voltage, Ifastcounterbalances Ileak and any furtherdepolarization will cause progressive activation ofIfast. (B) Changingḡfast in the 2D model hadequivalent effects on the shape of the steady stateI–V curves. Unlike in (A), voltage at theapex of the instantaneousI–V curve (purple arrows)changes as ḡfast is varied; inother words, the net current at perithreshold potentials can bemodulated by changing fast currents (which directly impact voltagethreshold) rather than by changing the amplitude or voltage-dependencyof slow currents. This is consistent with results in Figure 8. (C) Speedingup the kinetics of Islow impacts the onsetof class 2 and 3 excitability. Compared with original model(φw = 0.15;black), increasing φw to 0.25(red) increased Istim required to cause aHopf bifurcation or a QSC, but did not affectIstim required to cause an SNICbifurcation; reducing φw to 0.10(green) had the opposite effect (summarized in right panel). Increasingφw also widened thediscontinuity in the class 2 f–I curve andallowed class 2 and 3 neurons to achieve higher spiking rates withstrong Istim because of the faster recoverybetween spikes; reducing φw hadthe opposite effects.
Mentions: Interpretation of the phase plane geometry can be formalized by doing localstability analysis near the fixed points ([27], see also chapter11 in [28]). In class 3 neurons, at the stable fixed point. This means, at steadystate, that positive feedback is slower than the rate of negativefeedback,φw/τw.Subthreshold activation of Islow produces a steadystate I–V curve that is monotonic and sufficientlysteep near the apex of the instantaneous I–V curvethat V is prohibited from rising high enough to stronglyactivate Ifast (Figure 9A, left). However, because the twofeedback processes have different kinetics, a spike can be initiated if thesystem is perturbed from steady state: if V escapes high enoughto activate Ifast (e.g., at the onset of an abruptstep in Istim), fast-activating inward current canoverpower slow-activating outward current—the latter is stronger whenfully activated, but can only partially activate (because of its slow kinetics)before a spike is inevitable. Through this mechanism, a single spike can beinitiated before negative feedback forces the system back to its stable fixedpoint, hence class 3 excitability. Speeding up the kinetics ofIslow predictably allowsIslow to compete more effectively withIfast (see below).

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