The Journal of Neurophysiology Vol. 78 No. 5 November 1997, pp. 2616-2630
Copyright ©1997 by the American Physiological Society

Stochastic Threshold Characterization of the Intensity of Active Channel Dynamical Action Potential Generation

Robert M. Schmich and Michael I. Miller

Department of Electrical Engineering, Washington University, St. Louis, Missouri 63130

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Schmich, Robert M. and Michael I. Miller. Stochastic threshold characterization of the intensity of active channel dynamical action potential generation. J. Neurophysiol. 78: 2616-2630, 1997. This paper develops a stochastic intensity description for action potential generation formulated in terms of stochastic processes, which are direct analogues of the physiological processes of the pre- and postsynaptic complex of the cochlear nerve: 1) neurotransmitter release is modeled as an inhomogeneous Poisson counting process with release intensity µ_t, 2) the excitatory postsynaptic conductance (EPSC) process is modeled as a marked, linearly filtered Poisson process resulting from the linear superposition of standard shaped postsynaptic conductances of size G, and 3) action potential generation is modeled as resulting from the EPSC exceeding a random threshold determined by active channel dynamics of the Hodgkin-Huxley type. The random threshold is defined to be the least upper bound in the size of a standard-shaped neurotransmitter release injected at time t given the previous action potential time and the number of releases occurring in a short preconditioning time increment. The action potential process is modeled as a self-exciting point process with stochastic intensity resulting from the probability that the random threshold process crosses the threshold in some small time increment that is a function of time since previous action potential, release intensity, and the probability that a single synaptic event exceeds the stochastic threshold. The stochastic intensity model is consistent with a direct simulation of the nonlinear Hodgkin-Huxley differential equations over a variety of parameters for the vesicle release intensity, vesicle size, vesicle duration, and temperatures. Results are presented showing that the regularity properties seen in the vestibular primary afferent in the lizard, Calotes versicolor, associated with a slow-to-activate potassium channel resulting in a long afterhyperpolarization can be accommodated directly by the stochastic intensity description. The stimulus dependence of the model is attributed to synaptic transmission and the probabilistic nature to the threshold conductance process, which is dependent upon the EPSC process. The stochastic intensity is seen to have a form consistent with the phenomenologically based Siebert-Gaumond model, a stimulus-related function of time multiplied by a refractory-related function of time since previous action potential.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

This paper develops a stochastic intensity description for action potential generation in the cochlear nerve associated with active channel dynamics of the Hodgkin-Huxley-type. The model is formulated in terms of coupled stochastic processes that are direct analogues of the following physiological processes: 1) vesicle release is modeled as an inhomogeneous Poisson counting process {M_t, t  0} with random event times {u_i, i = 1, . . . , M_T} and release intensity µ_t = lim₀ Pr{M_t,t+ = 1}/, 2) excitatory postsynaptic conductance (EPSC) {G_t, t  0}, which is modeled as a marked, linearly filtered Poisson process consisting of a superposition of standard shaped neurotransmitter releases of size G, and 3) action potential generation is a counting process {N_t, t  0} with event times {w_i, i = 1, . . . , N_T}. These are depicted in the Fig. 1, left.

View larger version (18K): [in this window] [in a new window]   FIG. 1. Left: diagram showing the stochastic processes of the model: 1) synaptic vesicle release Mt an inhomogeneous Poisson counting process, 2) excitatory postsynaptic conductance (EPSC) Gt, a marked linearly filtered Poisson process, and 3) action potential generation Nt, resulting from the nonlinear Hodgkin-Huxley (H-H) dynamics. Right, top: Poisson transmitter release process Mt with the Poisson random occurrence times {ui, i = 1, . . . , MT} and the single transmitter EPSC process obtained by adjoining the single transmitter response to each random release time ui. Middle: EPSC process Gt resulting from the linear superposition of the standard-shaped transmitter release process. Bottom: action potential process Nt occurring on random times {wi, i = 1, . . . , NT}, resulting from the solution of the H-H dynamical channel model.

Our approach for modeling spike generation is to characterize the threshold properties of nonlinear active channel dynamics. We model action potential generation as resulting from the random EPSC process passing threshold for the nonlinear active channel dynamical systems of the Hodgkin-Huxley (H-H) type. Action potential generation is modeled as a self-exciting point process with intensity determined by the neurotransmitter release intensity µ_t, stochastic threshold, and synaptic conductance conditioning history. Fundamental to our model is the notion that action potential generation results from the random EPSC process driving the nonlinear channel dynamics to cross threshold on random times {w_i, i = 1, . . . , N_T}, which form the action potential process. To calculate the intensity of discharge, we define the random synaptic threshold conductance function (t  w_Nt = , m) as the just-necessary-sized neurotransmitter quanta injected at time t required to cause an action potential at time t as a function of the time since previous action potential t  w_Nt =  and the number of preconditioning vesicles M_tH,t = m occurring in a small history of the synaptic conductance process [t  H, t). By calculating the probability laws on the sequence of thresholds (, 0), (, 1), . . . , we can derive the stochastic intensity of discharge conditioned on the history of the action potential times. The parameters of the model are the temperature, vesicle duration, vesicle intensity and size, correlation, and channel characteristics of the postsynaptic membrane.

For characterizing a particular active channel system, we empirically calculate the stochastic threshold distribution Pr{(, m)  G} as a function of time since previous action potential  and preconditioning events m by adapting the channels and temperatures of a single compartmental H-H model to the system of interest and empirically calculating the stochastic threshold for that system. For simulating the threshold characteristics of the afferent terminal, excitatory synaptic currents, passive membrane components, and the voltage-gated currents generating the action potential are included. These are depicted in Fig. 2, left, which shows the idealized model with membrane capacitance in parallel with the sodium, potassium, leak, and synaptic conductances G^Na_t, G^K_t, G^L, and G_t, having reversal potentials V^Na, V^K, V^L, V^S, and postsynaptic membrane voltage V_t. Throughout the paper, a variety of stochastic threshold functions have been generated from the nonlinear H-H systems, each one reflecting the range of systems that are of current interest to us, including fish vestibular neurons at 17-27°C (Boyle and Highstein 1990; Rabbitt et al. 1994, 1995) as well as bird (Boyle et al. 1994) and, of course, mammalian auditory nerve at 37°C. As well, we present results that model the vestibular primary afferent in the lizard Calotes versicolor at 32°C as described by Highstein and coworkers (Schessel et al. 1991). For examining the regularity properties of neural discharge in the vestibular system, we include a slow-to-activate potassium channel causing long afterhyperpolarizations. H-H computation results are presented that illustrate excellent agreement between the stochastic intensity of discharge model and the H-H spike generation model.

View larger version (14K): [in this window] [in a new window]   FIG. 2. Left: idealized membrane patch model relating the EPSC process, Gt, and the postsynaptic membrane voltage, Vt. Middle: example solution of Eq. 1 (- - -) with the driving EPSC process t = {Gt, t  0} superimposed () at 17°C. Right: synaptic conductance threshold t(t  wNt, t) plotted as a function of time t during a conductance sample path t.

A strong departure of this work from previous work in Miller and Wang (1993) is the predominant emphasis on the stochastic intensity associated with time since previous action potential strictly larger than dead time. Herein we do not explore the nonzero probability behavior associated with the discontinuity in threshold as active channel systems release from dead time. As well, we parameterize threshold in terms of synaptic conductance directly, not requiring the small signal approximations assumed previously. This requires a characterization of the active channel systems in terms of a stochastic threshold measured in synaptic conductance units. As demonstrated, the stochastic synaptic conductance threshold determines the stochastic intensity of action potential generation. Interestingly, we show that the model is consistent with the product model of Siebert and Gray (Gray 1967; Siebert and Gray, 1963) and Gaumond, Kim, and Molnar (Gaumond et al. 1982, 1983). The stochastic intensity derived here has a natural factorization in terms of synaptic vesicle release intensity µ_t and the threshold characteristics of the postsynaptic active channel dynamical system having a clear functional dependence on the time since previous action potential. However, as discussed below, the stochastic intensity can be a nonlinear function of vesicle release intensity and can diverge from the linear model at varying stimulus levels.

	METHODS

Abstract Introduction Methods Results Discussion References

Stochastic Hodgkin-Huxley threshold characteristics

Neurotransmitter release is modeled as a Poisson counting process {M_t, t  0} with intensity µ_t = lim₀ Pr{M_t,t+ = 1}/. At this time, we take this as an ordinary inhomogeneous Poisson process; this has been extended as described in Wang et al. (1994a,b) and Schmich (Schmich 1996; Schmich and Miller 1997), including short-term adaptation reflected by the synaptic state. For our purposes here for accounting for steady state responses such as spontaneous activity, µ_t = µ is a fixed deterministic, unknown parameter. The resulting EPSC process, {G_t, t  0}, is modeled as a marked, linearly filtered Poisson process (Snyder and Miller 1991, chapts. 4 and 5) resulting from each neurotransmitter release adding an independent, short time increment of synaptic conductance of some random height in the postsynaptic cleft independent of others. Linear superposition of quantal conductances is assumed. The single transmitter response is idealized to a fixed waveform of standard shape (t), t  0. A response shifted to random time u_i with height G_i is denoted as G_iui (t). The synaptic conductance process {G_t, t  0}, due to a number of such conductance enhancements occurring on Poisson random times {u₁, . . . , u_MT}, becomes G_t =  G_iui(t). The exact expression for the characteristic function of the process G_t can be found in Snyder and Miller (1991), pg. 219, for arbitrary system response. We take the synaptic conductance to have fixed rectangular waveforms of duration W, (t) = 1, t  [0, W], and 0 otherwise, with _ui (t) = 1, t  [u_i, W + u_i], and 0 otherwise. The random heights are taken to be a fixed deterministic quantity of size G (see Miller and Wang 1993 for the more general formulation), resulting in the EPSC process becoming G_t = M_tW,tG. The construction of the EPSC process is shown in Fig. 1, right.

For calculating the random threshold properties associated with particular active channel dynamics, we use the standard H-H-type differential equation, supplemented with the standard kinetic equations for both G^Na_t and G^K_t, for describing the input and output relationship between postsynaptic conductance G_t and membrane voltage V_t

(1)

A typical solution is shown in Fig. 2, middle, using the standard method based on the Frankenhaeuser-Huxley dynamics (Frankenhaeuser and Huxley 1964) at 17°C. An EPSC sample path was constructed by quantal neurotransmitter releases on Poisson random times, adjoining a single transmitter response to each release time with {G_t, t  0} resulting from the linear superposition of all releases on the postsynaptic cleft. The EPSC process (solid) is the input to Eq. 1 with the postsynaptic membrane voltage V_t shown as the dashed line. For this temperature, it appears that a single synaptic conductance is the steady state threshold of the system. The system exhibits the well-known phenomena of absolute refractory periods, with dead time denoted by T_D, and relative refractory periods. A relative refractory period is shown clearly at t = 19 ms, with the two overlapping conductances producing a subthreshold response, because there was a previous spike at t = 15-16 ms. An absolute refractory period is exhibited by the system near t = 45 ms. Notice, there are three overlapping conductances during the falling phase of the action potential or absolute refractory period of the system and thus no new action potential is generated.

The action potential process {N_t, t  0} and its random times {w_i, i = 1, . . . , N_T} are generated by solving the stochastic Eq. 1 and defining the action potential times to be the time when the voltage waveform increases rapidly toward V_Na and obtains its peak value. These are depicted in the Fig. 1, bottom right.

To calculate the intensity of discharge of action potential generation as a function of time since previous spike t  w_Nt, we must be able to calculate to o(), Pr{N_t,t+ = 1|t w_Nt = }, where the random measure on the spike generation process is induced by the measure on the synaptic conductance processes {G_t, t  0}. For this, the H-H generation of action potentials is characterized as a nonlinear thresholder. The action potential times {w_i, i = 1, . . . , N_T} result from the random threshold crossing of the synaptic conductance process {G_t, t  0}.

Begin by defining threshold, a random process totally determined by the synaptic conductance sample paths. Define the random element _t  {G,  <  < t} a sample path of the EPSC process in the sample space _t = {_t = {G,  <  < t}} of conductance processes. Also define the subset of elements _t()  _t of sample paths having no action potentials in [t  , t), _t() = {_{t  t} : t  w_Nt = }.

The random threshold process { (t  w_Nt = , _t) t > } is taken as a function of _{t  t}(), the random sample conductances for which the time since previous action potential is t  w_Nt = . Definition 1: The random threshold function (, _t) at time t is defined as the just-necessary standard-shaped vesicle of size  injected at time t required to cause an action potential with preconditioning _{t  t}() where t  w_Nt = . See APPENDIX A and Eq. A3 for a formal definition.

Figure 2, right, shows an example of the threshold function computed throughout the conductance sample path _t = {G, 0   < t} with an action potential assumed to reset the system at time t = 0. Figure 2, right, shows (t  w_Nt, _t) computed from the H-H dynamical system according to Eq. A3 generated by solving the H-H differential equation at 17°C. Notice how the threshold resets dramatically after an action potential; also notice the absolute dead time T_D = 3.0 ms.

We have assumed implicitly that the threshold function (t  w_Nt, _t) is not an explicit function of time t only in its dependence on the time since previous action potential and on the conductance sample path. As well, we assume (t  w_Nt, _t) is continuous in its argument t, i.e., (t +   w_Nt+, _t+)  (t  w_Nt, _t) as   0. This assumes t  w_Nt > T_D, the absolute dead time! Not surprisingly, the threshold function is discontinuous at absolute dead time  = T_D. As shown in Miller and Wang (1993), this must be handled separately when describing stochastic intensities. Our analysis throughout assumes t  w_Nt > T_D so that continuity holds.

Notice that in defining the threshold function, we have conditioned explicitly on the time since previous action potential precisely because channel kinetics change dramatically as a function of time since previous action potential. This allows for the efficient encoding of the conditional intensity of discharge. This is obvious as shown in Fig. 2, right. The self-exciting point process conditioned on conductance sample paths becomes

(2)

where 1_[0,G][(t  w_Nt, _t)] is the set indicator function that equals one for (t  w_Nt, _t)  [0, G] and zero otherwise. As well, if G is a random size conductance, as in Miller and Wang (1993), then

(3)

We choose to define the stochastic intensity as a function of a finite history of synaptic conductance rather than infinite sample path history. For this, we assume the nonlinear action potential generation process is essentially a countable state Markov machine, the states corresponding to the number of synaptic events in a small preconditioning interval. The stochastic threshold at time t becomes a function only of M_tH,t, the number of preconditioning vesicles in time increment [t  H,t), and the time since previous action potential. The history parameter H is on the order of membrane integration for passive dynamics and intuitively will express the fact that H-H dynamics conditioned on being in an interval where there is no action potential generated [w_Nt, t) will be purely passive in its response with a time constant that does not reflect long-term history in the synaptic conductance sample path. As motivation, examine Fig. 3, which shows H-H thresholds that were computed to preconditioning stimuli at both 17°C (left) and 27°C (right) in which single conductances were released on various times from 5 ms, ₅(t) to 0.1 ms, _0.1(t). The H-H differential equation then was driven forward to t = 0 by solving Eq. 1 for the membrane voltage, {V,  < 0} with the associated synaptic conductance threshold {(  w_N, ),  < 0} calculated. The resulting threshold as a function of preconditioning transmitter release times are shown in Fig. 3. Quantal transmitter width for ₀(·) was chosen to be W = 2.0 ms (left) and W = 0.5 ms (right). Conductance history before W ms has a markedly decreased effect on the threshold of the system at time 0. 

View larger version (15K): [in this window] [in a new window]   FIG. 3. Synaptic conductance threshold plotted as a function of quantal transmittal release history. Quantal transmitter width W = 2.0 ms at 17°C (left); W = 0.5 ms at 27°C (right).

To examine the history parameter somewhat more quantitatively, we have computed the conditional covariance between the H-H action potential process {N_t, t  0} and the input synaptic conductance process {G_t, t  0} The cross-covariance is computed empirically over many trials where {w_i(), i = 1, . . . , N_T()} are the event times that satisfy the conditioning, w_i()    w_i()  t_Nwi(), and N_T() is the total number of events satisfying the conditioning. Figure 4 shows covariance curves as a function of correlation distance s at both 17°C for  = 4, 12, W = 1.0 ms (left) and 37°C for  = 2, 12, W = 0.5 ms (right) with transmitter release intensity, µ = 250 s. The covariance curves demonstrate that for correlation times s > W, the action potential process and the synaptic conductance process have small correlation. Synaptic conductance history {G,  (, t  W)} virtually is uncorrelated with the threshold conductance process at time t. Postsynaptic conductance history in the interval [t  H, t) has the greatest effect on the threshold. This use of the covariance provides a systematic way of choosing the preconditioning interval of size H. In general, channel dynamics can be expected to determine the covariance, slow systems will require larger correlation time. Shown in RESULTS is a modeled vestibular neuron with a long afterhyperpolarizing channel with longer correlation distances.

View larger version (16K): [in this window] [in a new window]   FIG. 4. Conditional covariance C(s|) between the synaptic conductance process Gt and the action potential process Nt plotted as a function of correlation distance s, conditioned on time  since previous action potential. Left: width W = 1.0 ms and release rate µ = 250 s at 17°C. Right: width W = 0.5 ms and release rate µ = 250 s at 37°C.

With the notion of equivalence defined through finite history, we compute the probability law on stochastic thresholds by dividing the sample paths into subsets _t(),   0, of sample paths for which there are no action potentials in [t  , t) with t  w_Nt = , and subsets _t(, m) of elements having M_tH,t = m vesicles released in [t  H,t) Notice, _t()   _t(, m). Definition 2: Define the family of random threshold processes, {(, m),  > T_D, m = 0, 1, 2, . . .}, to be the just-necessary conductance change of shape _t(·) required to elicit an action potential at time t, given  = t  w_Nt and M_tH,t = m or equivalently _{t  t}(, m).

Empirical generation of stochastic threshold

Our strategy is to characterize each active channel system by empirically computing the stochastic threshold from a nonlinear dynamical system with parameters chosen to reflect that of the system being characterized. The probabilistic and average properties of threshold can be calculated straightforwardly for subsets _t(, m), m = 0, 1, 2, . . . . Shown in Fig. 5 are the mean thresholds, denoted as (, m), characterizing the average threshold dynamics for the m = 0, 1, 2 subsets. The mean thresholds are calculated by initially causing the system to spike at time , driving the system forward to time 0 with M_H,0 = m synaptic conductances as the preconditioning stimulus and then calculating the conductance required at t = 0 to elicit an action potential by altering the height of a standard-shaped tansmitter release ₀(·). The preconditioning is varied over all Poisson release times for M_H,0 = m, m = 0, 1, 2. Figure 5, top, shows a pictorial representation for generating these characterizations. Notice, there is no randomness for them = 0 subset because there are no preconditioning stimuli in the preconditioning interval [H, 0). Figure 5, bottom, shows the associated H-H mean threshold functions (, m) for the m = 0, 1, 2 subsets as a function of the time since previous spike , with varying parameters for temperature (17, 27, and 37°C). The mean threshold curves (, m) illustrate the discharge history dependence in the model: during the absolute dead time interval, the mean threshold is effectively infinite, after which there is a decrease to the steady state threshold conductance with various time courses depending upon temperature and membrane currents. Also notice that both (, 1) at 27 and 37°C and (, 2) at 17, 27, and 37°C decrease to 0 because the preconditioning stimuli themselves are sufficient to generate action potentials.

View larger version (21K): [in this window] [in a new window]   FIG. 5. Top: empirical approach depicted for calculating the mean threshold (, m) for m = 0 (left), m = 1 (middle), and m = 2 (right) transmitter quanta injected as a preconditioning stimulus. Bottom: mean threshold conductance (, m) calculated by injecting transmitter on random times and computing the average necessary threshold according to Eq. 4 as a function of time t  wNt =  since previous action potential.

The H-H mean threshold function (, 0) shown in Fig. 5, bottom left, also explains the difference in the covariance curves seen in Fig. 4 for 17°C (left) and 27°C (right). Notice that the covariance curves for  = 4, 12 ms at 17°C are not equivalent, whereas the covariance curves for  = 2, 12 ms at 27°C are equivalent. This difference is explained by the shapes of the mean threshold curves shown in Fig. 5 with the threshold curve for 27°C reaching steady state level near  = 2 ms; the threshold curve for 17°C does not reach steady state until  = 13 ms. Thus the mean thresholds for  = 2, 12 ms at 27°C in Fig. 4 are equal, causing the respective covariance curves to be equivalent. For 17°C, the mean thresholds at  = 4, 12 ms are not equivalent, implying the respective covariance curves will be different.

Stochastic intensity of discharge

To study the global probabilistic behavior of the action potential process, it should be described via its intensity, which is the local probability of discharging in a small increment conditioned on the history of the process (chapter 6, Snyder and Miller 1991) and is given by

(4)

We calculate the local probability of discharge in a small time increment, [t,t + ] conditioned on the time since previous spike, t  w_Nt = , and the conductance sample paths associated with M_tH,t = m, m = 0, 1, 2, . . . vesicles released in [t  H,t] The first equation is the probability of discharge to o() conditioned on M_tH,t = m = 0 preconditioning transmitter quanta, which is the probability of release in [t, t + ], µ_t, multiplied by the probability that the associated threshold function, (, 0), is less than the standard size release G. The second equation is the probability of discharge to o() conditioned on there being one release in [t  H,t) and t  w_Nt = . This also applies similarly forM_tH,t = m = 2, 3, . . . . Notice, the stochastic intensity requires the probability that (, m) is less than a standard-shaped release injected at time t, Pr{(, m)  G}.

To calculate conditional probabilities over waveforms containing m vesicles with no action potentials N_t,t = 0 for time duration  = t  w_Nt, define (, m)  [0, 1] to be the probability of the subset _t(, m) conditioned on _t(m)

(5)

The final form for the intensity of discharge is obtained by using Bayes law resulting from the superposition of all events corresponding to m = 0, 1, 2, . . . transmitter release in [t  H,t) weighted by the Poisson law for vesicle release. This provides the following theorem. Theorem 1: Given is 1)  = t  w_Nt > T_D absolute dead time, 2) the set of probability distributions Pr{(, m)  G} characterizing the random threshold processes {(, m),   0}, m = 0, 1, . . . the probability of a single synaptic event size G causing an action potential given M_tH,t = m preconditioning neurotransmitter releases, and 3) the relative probabilities of each preconditioning subset Then the stochastic intensity of action potential generation becomes

(6)

Proof: see APPENDIX B.

Equation 6 is a stochastic intensity description of action potential generation associated with active channel dynamics of H-H type models. Essentially, this is the superposition of all events associated with m = 0, 1, 2, . . . preconditioning transmitter quanta, multiplied by the probability of one synaptic event in [t, t + ] causing an action potential, which goes as µ_t Pr{(, m)  G}.

Figure 6 shows the calculation of the probabilities Pr{(, m)  G} as a function of  for m = 1, 2, 3, . . . . Note that because there is no randomness in the preconditioning interval for (, 0), Pr{(, 0)  G} 1 if (, 0)  G, and 0 otherwise. For the mth subset, the probability law is calculated for the just-necessary conductance required to elicit an action potential at time t. The exact location of the m preconditioned vesicles in [t  H,t) is varied according to the Poisson occurrence times, with the probability curves Pr{(, m)  G} computed for m = 1, 2, 3, . . . as a function of time since previous spike  by varying the placement of the preconditioning transmitter quanta in [t  H,t). These curves were generated by uniformly releasing m transmitter quanta in the interval [t  H,t) over n trials and empirically calculating the associated probability function Pr{(, m)  G} for each ,

(7)

where 1_[0,G][_i(, m)] is the set indicator function equaling 1 for _i(, m)  [0,G], and 0 otherwise. Shown in Fig. 6, top, are plots of the probability curves as a function of  for the 17°C temperature. For the generated curves, 200 trials were calculated for each choice M_tH,t = m preconditioning vesicles, placing m events on uniform random times in [t  H,t). The threshold function (, m) decreases as m increases, causing the probability to rise to 1 quickly. Notice the probability curves increase for  > T_D because the threshold decreases monotonically with time since previous action potential . The parameter (, m) = Pr{N_t,t = 0|M_tH,t = m}, which is the percentage of waveforms not giving rise to an action potential in [w_Nt, t), is shown in Fig. 6, bottom, for m = 1, 2, 3. These curves were generated using the same parameters for the probability curves in Fig. 6, top. Notice, as the threshold decreases with , the number of sample paths that do not cause spiking in the preconditioning interval decreases to 0 dramatically: (, m) = Pr{t  w_Nt = |M_tH,t = m}0 as m grows. Also, notice that in Fig. 6, top, panel 3, the probability curve for m = 3 goes to 0 at  > 4 ms because the associated threshold function is negative, i.e. (, 3) < 0, which means that three neurotransmitter quanta in the preconditioning interval is sufficient to cause an action potential for  > 4 ms. Notice as well the curve ( > 4, m = 3) = 0. We define the probability over these events to be 0 as they do not enter into the calculation of the stochastic intensity.

View larger version (19K): [in this window] [in a new window]   FIG. 6. Top: family shown of probability curves Pr{(, m)  G = 6.8 mS/cm2}, m = 1, 2, 3, as a function of time since previous spike  at 17°C. Bottom: (, m) = Pr{t  wNt = |Mth,t = m}, the percentage of conductance waveforms that do not give rise to an action potential in [t  , t).

	RESULTS

Abstract Introduction Methods Results Discussion References

Stochastic intensity model of H-H dynamics

We use hazard rate histograms to compare the accuracy of the stochastic intensity model, _t() of Theorem 1, to the output of the actual H-H dynamical system driven by the EPSC process. Hazard rate functions (chapter 6, Snyder and Miller 1991) characterize the discharge intensity for such self-exciting point processes and have the following functional form: h[i] = (n[i])/(N_T  n[i]), where n[i] is the number of interarrivals in the ith bin and N_T is the total number of action potentials. This form for the hazard rate function assumes that the discharge intensity is stationary, i.e., _t() = (), or equivalently that the stimulus function is stationary, µ_t = µ. The general nonstationary form for the hazard rate function for _t() is derived in Mark and Miller (1992). The hazard function is essentially the maximum likelihood estimate of the intensity () with stimulus µ_t = µ (Mark and Miller 1992). The simulation results are generated assuming a stationary form for the stimulus function and, subsequently, the discharge intensity.

Begin by comparing the stochastic intensity description and the H-H active channel model using parameters from the classic paper of Frankenhaueser and Huxley (1964) at 20°C. Figure 7, left, shows the ionic currents resulting from the current stimulation at 20°C corresponding to Frankenhaueser and Huxley (1964), their Fig. 6. Figure 7 (right, dotted line) shows the hazard discharge rate generated from solving the differential Eq. 1 for action potential generation and computing h[i] = (n[i])/(N_T  n[i]). Superimposed via the solid line is the stochastic intensity () as a function of , the time since previous action potential. Notice the accurate fit of the stochastic intensity model of Theorem 1 with the H-H generated data. The dead time parameter T_D is replicated exactly; the rapid rise in hazard intensity as well as the neural discharge passing into its relative refractory period are all duplicated as well.

View larger version (18K): [in this window] [in a new window]   FIG. 7. Left: reproduction of Fig. 6 in Frankenhaueser and Huxley (1964), showing the computed ionic currents resulting from the same current stimulation of 1 mA/cm2 for 0 < t < 0.12 ms at 20°C. Right: plots of H-H hazard rate obtained by solving the H-H differential equation (···) and stochastic intensity model () () as a function of time since previous action potential  at 20°C, with µ = 300 s G = 7.86 mS/cm2, and W = 0.7 ms.

Figure 8 shows a comparison of the stochastic intensity model () () and the H-H dynamical equation's hazard rates (···) at 17°C. The vesicle size was increased fromG = 6.8 mS/cm² (Fig. 8, top left) to G = 9 mS/cm² (top right), and the release intensity was increased from µ = 200 s (bottom left) to µ = 300 s (bottom right). The single transmitter width was W = 0.5 ms (top) and W = 1.0 ms (bottom). Conditioning history was selected to be the vesicle width H = W in all four plots. Notice that the firing rate increases with transmitter quantal size G (top).

View larger version (36K): [in this window] [in a new window]   FIG. 8. Plots of hazard rates obtained by solving the H-H differential equation at 17°C (···) and () () as a function of . Top left: µ = 300 s, G = 6.8 mS/cm2, H = W = 0.5 ms. Top right: µ = 300 s, G = 9 mS/cm2, H = W = 0.5 ms. Bottom left: µ = 200 s, G = 7.5 mS/cm2, H = W = 1.0 ms. Bottom right: µ = 300 s, G = 7.5 mS/cm2, H = W = 1.0 ms.

Shown in Fig. 9 are the discharge hazard rates generated from solving the H-H model at 27°C (···) as a function of time since previous action potential. Shown superimposed () is the stochastic intensity (). The stochastic models are shown with transmitter quantal size G increased from 6 to 8 in Fig. 9, top, and transmitter release intensity µ decreased from 500 to 250 s from the left to right panels in Fig. 9, bottom. The width, W = 0.5 ms in Fig. 9, top, was decreased to 0.3 ms in Fig. 9, bottom. The preconditioning interval H was selected to be the transmitter width H = W in all four plots.

View larger version (38K): [in this window] [in a new window]   FIG. 9. Plots of hazard rates obtained by solving the H-H differential equation (···) and stochastic intensity () () as a function of time since previous action potential  at 27°C. Top left: µ = 500 s, G = 6 mS/cm2, H = W = 0.5 ms. Top right: µ = 500 s, G = 8 mS/cm2, H = W = 0.5 ms. Bottom left: µ = 500 s, G = 8 mS/cm2, H = W = 0.3 ms. Bottom right: µ = 250 s, G = 8 mS/cm2, H = W = 0.3 ms.

Figure 10 examines the rapid dynamics of high-temperature systems such as is appropriate for mammalian auditory nerve. The stochastic intensity, () () and hazard rate intensities (···) are shown at 37°C with µ = 250 s (top left), µ = 375 s (top right), µ = 500 s (bottom left), and µ = 625 s (bottom right), with H = W = 0.5 ms. Notice, that the discharge rates for 37°C almost immediately reach their steady state level for  > T_D  0.6 ms. This is consistent with the mean threshold curve dynamics (, m) shown in Fig. 5 for the 37°C curves (solid line).

View larger version (40K): [in this window] [in a new window]   FIG. 10. Plots of H-H hazard rates obtained by solving the H-H differential equation (···) and stochastic intensity model () () as a function of time since previous action potential  at 37°C. Top left: µ = 250 s, G = 7 mS/cm2, H = W = 0.5 ms. Top right: µ = 375 s, G = 7 mS/cm2,