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Conceived and designed the experiments: AJS MLL OYH. Analyzed the data: AJS. Contributed reagents/materials/analysis tools: AJS MLL. Wrote the paper: AJS MLL. Performed simulations: AJS. Coordinated study: MLL OYH.

The authors have declared that no competing interests exist.

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.

Computational modeling is of importance in striving to understand the complex dynamic behavior of a neuron. In neuronal modeling, the function of the neuron's components, including the cell membrane and voltage-dependent ion channels, is typically described using deterministic ordinary differential equations that always provide the same model output when repeating computer simulations with fixed model parameter values. It is well known, however, that the behavior of neurons and voltage-dependent ion channels is stochastic in nature. A stochastic modeling approach based on probabilistically describing the transition rates of ion channels has therefore gained interest due to its ability to produce more accurate results than the deterministic approaches. These Markov chain type of models are, however, relatively time-consuming to simulate. Thus it is important to develop new modeling and simulation approaches that take into account the stochasticity inherently present in the function of ion channels. In this study, we seek new stochastic methods for modeling the dynamic behavior of neurons. We apply stochastic differential equations (SDEs) and Brownian motion that are also commonly used in the air space industry and in economics. An SDE is a differential equation in which one or more of the terms of the mathematical equation are stochastic processes. Computer simulations show that the irregular firing behavior of a small neuron, in our case the cerebellar granule cell, is reproduced more accurately in comparison to previous deterministic models. Furthermore, the computation is performed in a relatively fast manner compared to previous stochastic approaches. Additionally, the SDE method provides more sophisticated mathematical analysis tools compared to other, similar kinds of stochastic approaches. In the future, the new SDE model of the cerebellar granule cell can be used in studying the emergent behavior of cerebellar neural network circuitry.

Neurons express intrinsic bioelectrical activity which is known to be stochastic in nature. In order to understand this complex dynamic behavior, computational modeling is inevitable. But, how to develop models that are capable of mimicking the intrinsic dynamic behavior of the biological counterpart accurately? On the other hand, how can detailed models, possibly also incorporating some sort of stochasticity, be simulated in a reasonable time? These questions are crucial in creating computer models of neurons with better predictive capabilities.

It is well known that many components of a neuron and its membrane, including voltage-dependent ion channels, are essential for the dynamic behavior (see, e.g.,

Several stochastic approaches have previously been developed for modeling the bioelectrical activity of neurons and excitable tissue. Monte Carlo simulations using discrete Markov chain type of models have been performed to understand the role of randomly opening ion channels (so called microscopic approach;

Recent theoretical work has provided evidence that more emphasis should be put on ion channel stochasticity and its role in intrinsic dynamic behavior of neurons

The well-established H-H formalism has formed, to a great extent, the basis for building biophysically and anatomically detailed models of neurons. Subsequently, the roles of conductances (and, ion channels) have been addressed using these models. It should be noted, however, that the deterministic H-H formalism does not take into account the fact that the behavior of ion channels underlying the whole-cell ionic currents is stochastic in nature. In other words, the ion channel stochasticity has been ignored, as also pointed out by White et al.

In this study, we use cerebellar granule cell as a test case and examine how the behavior of a small-size neuron is altered when stochasticity is introduced into the deterministic H-H type of model. In short, granule cells are glutamatergic excitatory neurons which translate the mossy fiber input into parallel fiber input to Purkinje cells

Several deterministic models have been presented for the cerebellar granule cell during the past few years

In this study, we select to use the deterministic model of _{F}, K_{Dr}, K_{A}, K_{ir}, Ca_{HVA}, and BK_{Ca}; Na_{F} stands for the fast inactivating sodium channel, K_{Dr} for the delayed rectifier potassium channel, K_{A} for the transient A-type potassium channel, K_{ir} for the inward rectifier potassium channel, Ca_{HVA} for the high-voltage-activated calcium channel, and BK_{Ca} for the large-conductance calcium- and voltage-activated potassium channel) and simple calcium dynamics to describe the changes in the membrane potential. The model is based on the theory of equivalent electrical circuits, as is conventionally done in neuronal compartmental modeling. The change in membrane potential, _{m}_{app}(_{F} channels, we have_{F} channels, _{F} channels, respectively. Furthermore, constants ^{+}. The processing of calcium ions is assumed to take place in small volume close to cell membrane and is linked to BK_{Ca} channel function. The change in intracellular calcium concentration, [Ca^{2+}], is described by_{HVA} channels and ^{2+}]_{rest}, and τ_{Ca}

Constant | Value | Description |

_{m} | 0.57 Ωm^{2} | membrane resistance |

_{m} | 0.03 F/m^{2} | membrane capacitance |

_{m} | −0.025 V | equilibrium membrane potential |

+0.07 V | equilibrium potential for Na^{+} | |

−0.075 V | equilibrium potential for K^{+} | |

+0.14 V | equilibrium potential for Ca^{2+} | |

−0.085 V | equilibrium potential for _{Ca} | |

5.2⋅10^{−6} mol/C | constant for Ca^{2+} transfer into the cell | |

[Ca^{2+}]_{rest} | 100⋅10^{−6} mol/m^{3} | [Ca^{2+}] at rest |

τ_{Ca} | 1⋅10^{−3} s | time constant for the decay of |

intracellular free calcium | ||

dcell | 6⋅10^{−6} m | diameter of the granule cell |

dshell | 1⋅10^{−7} m | diameter of the shell defining the volume |

in which calcium ions are processed | ||

400 S/m^{2} | maximal conductance for Na_{F} | |

120 S/m^{2} | maximal conductance for K_{Dr} | |

10 S/m^{2} | maximal conductance for K_{A} | |

28 S/m^{2} | maximal conductance for K_{ir} | |

4.6 S/m^{2} | maximal conductance for Ca_{HVA} | |

30 S/m^{2} | maximal conductance for BK_{Ca} | |

3 | exponential for Na_{F} activation | |

1 | exponential for Na_{F} inactivation | |

4 | exponential for K_{Dr} activation | |

3 | exponential for K_{A} activation | |

1 | exponential for K_{A} inactivation | |

1 | exponential for K_{ir} activation | |

2 | exponential for Ca_{HVA} activation | |

1 | exponential for Ca_{HVA} inactivation | |

1 | exponential for BK_{Ca} activation |

See the sections Deterministic Model and Complete Stochastic Model for more details on ion channel types and the description of the complete mathematical model.

The parameter values of the original deterministic model have been selected based on data taken from in vivo and in vitro experimental records (for references see

The stochastic model is simulated for 0.5 seconds and depolarized from 0.15 seconds to 0.35 seconds. The value of the parameter σ is set to 0.15 and all other parameters are fixed as explained in the text.

In each trace, firing is simulated for 0.4 seconds with a time step of 10^{−5} seconds. In the upper panel the depolarizing current is below firing threshold (_{app} = _{app}_{app}

Depolarizing current pulses from 0 pA to 45 pA are used. For each value of depolarizing current we simulated 50 realizations, each 50 seconds long. Median, upper and lower quartiles, and the maximal and minimal firing frequencies are given for each depolarizing current pulse; outliers are marked with+symbol. Spontaneous activity is observed at low firing frequencies with depolarizing currents below 11 pA which is the firing threshold of the model. The

The random nature of synaptic activity, including the probabilistic release of neurotransmitters from synaptic vesicles, is one of the main sources of noise causing variability of firing. When modeling neuronal dynamics, stochasticity has thus been typically incorporated in the model input (see, e.g.,

There are alternative ways of introducing stochasticity in the behavior of the voltage-gated ion channels. In this work, we approximate the randomness in the operation of voltage-dependent ion channels as Brownian motion, i.e., as a Gaussian process with independent increments. Therefore, we convert the complete deterministic model for the cerebellar granule cell into a stochastic model. We describe the activation and inactivation of the six different ionic conductances using stochastic differential equations of the form

Here, the original deterministic equation _{X}_{X}

Using the common alternative notation, Equation 4 could also be given in the form

Specific types of autocorrelation functions have been used to avoid values for the gating variables which are not in the interval [0,1]. Autocorrelation function has been constructed so that it decreases the variance of the stochastic component when the value of a gating variable approaches 0 or 1. Although this approach decreases the probability of obtaining values outside the desired range, there is still a possibility that in a given point the realization of the stochastic component results in a value of the gating variable not in the interval [0,1].

It is possible to completely avoid values for the gating variables which are not in the interval [0,1]. The use of reflecting boundaries (i.e., the values under 0 or over 1 are reflected back to interval [0,1]) prevents the undesired values, but results in a model which does not correspond to the original stochastic integral equation (Equation 5).

In our model, we use a constant parameter σ and increments of Brownian motion, which ensures that the produced realizations are truly solutions of the corresponding integral equation. Another reason for selecting a constant parameter σ to our model is that, in the future, we are able to estimate its value using maximum likelihood estimation methods. This kind of estimation would be more difficult for a time-dependent parameter σ.

We have to be concerned about the undesired values of the gating variables, because the stochastic component in our model has now constant variance. This would result in problems when the values of the gating variables are close to 0 or 1. However, we are able to almost completely avoid undesired values for the gating variables by properly controlling the value of parameter σ. During depolarization only the gating variable for the K_{A} channel inactivation approaches zero and large negative values of the stochastic component would result in negative values of the gating variable. Hence, we have to use small values of parameter σ or use a separate parameter describing the stochastic fluctuations in the K_{A} channel inactivation process. For this paper, we choose the former approach and use the same, small value of parameter σ for all activation and inactivation processes. When the model is not depolarized, some of the gating variables are fluctuating relatively close to zero or one. This also limits our choice of proper value for the parameter σ.

In _{A} activation and inactivation process. From

The complete stochastic model used in this work is described with Equation 8. We use our independently developed simulation software in the MATLAB programming environment to make the calculations. The random numbers required in the simulations are generated with MATLAB's random number generators. The following equations are used to calculate the change in membrane potential, _{m}^{2+}], and in the gating variables for activation and inactivation processes at each time point

Channel | Process | Forward rate function | Backward rate function |

Na_{F} | activation | α_{NaF,a}(V_{m}) = 3•10^{3}•_{m}−0.01)+39•10^{−3})•0.081•10^{3} | β_{NaF,a}(V_{m}) = 3•10^{3}•_{m}−0.01)+39•10^{−3})•−0.066•10^{3} |

Na_{F} | inactivation | α_{NaF,i}(V_{m}) = 0.24•10^{3}•_{m}−0.01)+50•10^{−3})•−0.089•10^{3} | β_{NaF,i}(V_{m}) = 0.24•10^{3}•_{m}−0.01)+50•10^{−3})•0.089•10^{3} |

K_{Dr} | activation | α_{KDr,a}(V_{m}) = 0.34•10^{3}•_{m}−0.01)+38•10^{−3})•0.073•10^{3} | β_{KDr,a}(V_{m}) = 0.34•10^{3}•_{m}−0.01)+38•10^{−3})•−0.018•10^{3} |

K_{A} | activation | α_{KA,a}(V_{m}) = 2.2•10^{3}•_{m}−0.01)+46.7•10^{−3})•0.04•10^{3} | β_{KA,a}(V_{m}) = 2.2•10^{3}•_{m}−0.01)+46.7•10^{−3})•−0.01•10^{3} |

K_{A} | inactivation | α_{KA,i}(V_{m}) = 0.016•10^{3}•_{m}−0.01)+78.8•10^{−3})•−0.075•10^{3} | β_{KA,i}(V_{m}) = 0.016•10^{3}•_{m}−0.01)+78.8•10^{−3})•0.055•10^{3} |

K_{ir} | activation | α_{Kir,a}(V_{m}) = 0.133•10^{3}•_{m}−0.01)+83.94•10^{−3})•−0.0411•10^{3} | β_{Kir,a}(V_{m}) = 0.17•10^{3}•_{m}−0.01)+83.94•10^{−3})•0.028•10^{3} |

Ca_{HVA} | activation | α_{CaHVA,a}(V_{m}) = 0.049•10^{3}•_{m}−0.01)+29.06•10^{−3})•0.063•10^{3} | β_{CaHVA,a}(V_{m}) = 0.082•10^{3}•_{m}−0.01)+18.66•10^{−3})•−0.039•10^{3} |

Ca_{HVA} | inactivation | α_{CaHVA,i}(V_{m}) = 0.0013•10^{3}•_{m}−0.01)+48•10^{−3})•−0.055•10^{3} | β_{CaHVA,i}(V_{m}) = 0.0013•10^{3}•_{m}−0.01)+48•10^{−3})•0.012•10^{3} |

BK_{Ca} | activation | α_{BKCa,a}(V_{m},[Ca^{2+}]) = (2.5•10^{3})/(1+1.5•10^{−3}•^{3}•(V_{m}−0.01))/[Ca^{2+}]) | β_{BKCa,a}(V_{m},[Ca^{2+}]) = (1.5•10^{3})/(1+[Ca^{2+}]/(150•10^{−6} •^{3}•(V_{m}−0.01)))) |

In the model, _{i}_{i}_{i}_{i}_{i}_{i}_{i}

In stochastic simulation, we use the same parameter values as for the original deterministic model (_{i}_{i}_{i}^{−5} s.

Using this stochastic H-H type of model (see Equation 8), we are able to simulate, by intrinsic properties of the model, the following dynamic behavior

In the simulations, we observe

The highest firing rate the models can attain is approximately 300 Hz. Firing frequencies of up to 250 Hz have been observed with little or no adaptation of firing in response to strong depolarizing current pulses in in vivo granule cells _{A} effect (_{A} current shown to exist in in vitro granule cells

Experimental findings have indicated that irregularities in the firing of cerebellar granule cells are at least partly driven by intrinsic mechanisms, not exclusively by synaptic mechanisms. Irregularity in firing, as well as random subthreshold membrane oscillations, have been measured in the presence of 10 µM bicuculline blocking GABA-ergic inhibition

As an improvement to the deterministic granule cell model considered in this work

A small depolarizing current pulse (shown by a rectangular bar at the bottom of the figure) below firing threshold is injected into the cell soma. The bursts are evoked by random changes of σ between the values σ = 0.3 and σ = 1.1 (i.e., during a burst the value of parameter σ is increased to 1.1 otherwise it being 0.3). For illustrative purposes the trace with two bursts of action potentials is shown here (compare also with

Occasional

A small depolarizing current below firing threshold is applied throughout the simulation, similarly as in

A comparison between the responses obtained by the deterministic and stochastic model is shown in

The length of each trace is 0.4 seconds. In A) the depolarizing current is below firing threshold (_{app}_{app}_{app} = _{app}

A depolarizing current pulse just above the firing threshold is given. A simulation of 5 seconds is shown to provide evidence that stable solutions are obtained with stochastic differential equations and Brownian motion. The simulation time of this trace with time-step of 10^{−5} seconds is ca. 15 seconds. For this simulation σ = 0.5.

Variability in the firing caused by the parameter σ can be assessed by examining the histograms of interspike intervals with different values of depolarizing current pulses and different values of parameter σ (

Firing is simulated for 50 seconds with each depolarizing current pulse, _{app}_{app}_{app}_{app}

The existence of spontaneous firing can also be observed from

The coefficient of variation (CV) of the interspike intervals is often employed to quantify the regularity/irregularity of action potential firing. A completely regular firing has a CV of zero. In this work, the dependence of CV on different values of parameter σ and different depolarizing current pulses is studied. For the parameter values of σ = 0.1, 0.3, and 0.5, the results obtained for the mean, standard deviation (std), and CV are given in _{app}_{app}_{app}

Depolarizing current | σ | mean (s) | std (s) | CV |

_{app} | 0.1 | 0.0536 | 0.0461 | 0.8598 |

0.3 | 0.0251 | 0.0155 | 0.6194 | |

0.5 | 0.0205 | 0.0124 | 0.6055 | |

_{app} | 0.1 | 0.0247 | 0.0130 | 0.5282 |

0.3 | 0.0208 | 0.0118 | 0.5655 | |

0.5 | 0.0184 | 0.0106 | 0.5794 | |

_{app} | 0.1 | 0.0036 | 4.53⋅10^{−5} | 0.0125 |

0.3 | 0.0036 | 1.24⋅10^{−4} | 0.0343 | |

0.5 | 0.0036 | 2.04⋅10^{−4} | 0.0562 |

Firing is simulated for 50 seconds with three different values of depolarizing current pulses, _{app}_{app}_{th}_{app}_{app}

Bursts of action potentials have been recently recorded in in vivo granule cells in response to sensory stimuli using patch-clamp technique (cf.

The

Three sets of ten realizations of firing are simulated with the stochastic granule cell model using a depolarizing current pulse just above the firing threshold (_{app}

Based on the simulation results presented in the last four sections, it can be concluded that our new stochastic model is capable of reproducing the details of the firing shown for granule cells in vitro

In addition to putting emphasis on choosing the correct noise model, there is a need to consider computational efficiency, especially with realistic neuron models. Using the same simulation environment, the computation time of the SDE model is only two times the computation time of the deterministic model. In other words, the simulations of the SDE model can be run in a time-scale of seconds with a standard desktop PC (in our simulations, 1.86 GHz processor with 2 GB of RAM). For example, simulating a five-second trace for

We have shown here that, by using stochastic differential equations and Brownian motion to incorporate ion channel stochasticity, it is possible to reproduce with high precision the intrinsic electrophysiological activity of a neuron. The method presented here has several advantages over deterministic and other stochastic approaches. First, the approach provides models of neurons with realistic irregular behavior better than the deterministic approaches commonly used in computational neuroscience. Second, it decreases the computation time in comparison to discrete stochastic approaches. Additionally, the method provides more sophisticated mathematical analysis tools compared to other, continuous stochastic approaches. In the following, we discuss these advantages as well as the limitations of the proposed method and point out some possible extensions for future work.

In general, there are a number of ways to improve deterministic compartmental models and to make them more accurate and realistic, as has also been pointed out by Carelli et al.

As there are experimental findings showing that irregular behavior observed in an in vitro granule cell may be driven by intrinsic mechanisms (

Although several stochastic methods have been presented for describing the intrinsic activity of neurons (for a review, see, e.g.,

The computationally fast, yet accurate SDE model of the granule cell could be useful in studying the emergent behavior of cerebellar neural network circuitry. There are several interesting, experimentally observed phenomena that have to be addressed in the future, including the low-frequency oscillations observed in the cerebellar granule cell layer of awake, freely behaving rats

In addition to accurate reproduction of experimental findings, it is important to consider the computation time required by a specific stochastic approach. In many cases, lack of computing resources has prevented the use of stochasticity in detailed compartmental modeling. Moreover, there are very few studies reporting actual computation times to benchmark existing stochastic methods and to guide the selection of suitable method. Carelli et al.

The computation time of our SDE model is, in contrast, only two times the computation time of the deterministic model. Therefore, the computation time is considerably decreased in comparison to discrete-state stochastic approaches in which the ion channels' transition rates are described as discrete-state Markov processes. The SDE method thus makes it possible to simulate long time series, similarly as in

One advantage of the SDE approach is that the approach provides more sophisticated theoretical tools for analysis of models in comparison to other previously presented continuous-state stochastic approaches (see, e.g.,

SMC simulation based ML estimation is a Bayesian type of estimation technique which relies on transforming the probability distributions of the estimation problem into distributions which are easy to sample. This transformation allows us to use SMC approach when drawing samples from the desired posterior distributions. Based on these samples, a maximum-likelihood estimation technique is utilized for producing ML estimates for the selected model parameters. As an example, these parameters can include maximal conductances of ionic currents and the intensity of random fluctuations in the current-clamp data. This kind of fitting makes it possible to use irregular learning data in the estimation. Our ongoing work using the SDE version of the H-H model for a squid axon has shown that accurate ML estimates can be obtained for the selected model parameters based on irregular learning data

The SDE approach, inevitably, has certain challenges that need to be addressed in the future. First, the gating variables of the H-H type of models may have undesired values if no attention is paid to the selection of the value for the parameter σ. This problem may be corrected by implementing stochasticity into gating variables in such a way that the level of fluctuations is dependent on the value of the gating variable. This way we would be able to decrease the fluctuations when the value of the gating variable is approaching 0 or 1 thus decreasing the probability of obtaining values not in the interval [0,1]. This approach is, however, a matter of a future study. Second, none of the freely available neural simulation tools include the possibility to use stochastic differential equations. Presently, self-made simulation software is required which may hinder the use of SDEs in compartmental modeling. Inclusion of a variety of deterministic and stochastic methods in the simulation tools would greatly benefit neuroscientists in simulating the functions of neurons and, ultimately, of neural networks.

In the future, more work will be needed to clarify the roles of different types of noise sources for small, intermediate-size, and large-size neurons, both from experimental and theoretical points of view. As an example, when studying the effects of synaptic input noise the response dynamics of a nerve has been shown to be sensitive to the details of noise model

_{A}

^{+}-dependent mechanism.