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Conceived and designed the experiments: RCC CO MFN. Performed the experiments: RCC CO MFN. Analyzed the data: RCC CO MFN. Contributed reagents/materials/analysis tools: RCC MFN. Wrote the paper: MFN. Designed and developed simulation tools: RCC.

The authors have declared that no competing interests exist.

Neuronal activity is mediated through changes in the probability of stochastic transitions between open and closed states of ion channels. While differences in morphology define neuronal cell types and may underlie neurological disorders, very little is known about influences of stochastic ion channel gating in neurons with complex morphology. We introduce and validate new computational tools that enable efficient generation and simulation of models containing stochastic ion channels distributed across dendritic and axonal membranes. Comparison of five morphologically distinct neuronal cell types reveals that when all simulated neurons contain identical densities of stochastic ion channels, the amplitude of stochastic membrane potential fluctuations differs between cell types and depends on sub-cellular location. For typical neurons, the amplitude of membrane potential fluctuations depends on channel kinetics as well as open probability. Using a detailed model of a hippocampal CA1 pyramidal neuron, we show that when intrinsic ion channels gate stochastically, the probability of initiation of dendritic or somatic spikes by dendritic synaptic input varies continuously between zero and one, whereas when ion channels gate deterministically, the probability is either zero or one. At physiological firing rates, stochastic gating of dendritic ion channels almost completely accounts for probabilistic somatic and dendritic spikes generated by the fully stochastic model. These results suggest that the consequences of stochastic ion channel gating differ globally between neuronal cell-types and locally between neuronal compartments. Whereas dendritic neurons are often assumed to behave deterministically, our simulations suggest that a direct consequence of stochastic gating of intrinsic ion channels is that spike output may instead be a probabilistic function of patterns of synaptic input to dendrites.

The activity of neurons in the brain is mediated through changes in the probability of random transitions between open and closed states of ion channels. Since differences in morphology define distinct types of neuron and may underlie neurological disorders, it is important to understand how morphology influences the functional consequences of these random transitions. However, the complexities of neuronal morphology, together with the large number of ion channels expressed by a single neuron, have made this issue difficult to explore systematically. We introduce and validate new computational tools that enable efficient generation and simulation of models containing ion channels distributed across complex neuronal morphologies. Using these tools we demonstrate that the impact of random ion channel opening depends on neuronal morphology and ion channel kinetics. We show that in a realistic model of a neuron important for navigation and memory random gating of ion channels substantially modifies responses to synaptic input. Our results suggest a new and general perspective, whereby output from a neuron is a probabilistic rather than a fixed function of synaptic input to its dendrites. These results and new tools will contribute to the understanding of how intrinsic properties of neurons influence computations carried out within the brain.

The appropriate level of physical detail required to understand how complex processes such as cognition and behavior emerge from more simple biological structures is unclear

Investigation of stochastic ion channel gating using numerical simulations has been limited by trades-offs between simulation accuracy and computation time

To address the functional consequences of stochastic ion channel gating in neurons with extensive dendritic or axonal arborizations we developed a parallel stochastic ion channel simulator (PSICS), which enables efficient simulation of the electrical activity of neurons with complex morphologies and arbitrary localization of stochastic ion channels on the extracellular membrane, while also addressing limitations of previous approaches. We have also developed an interactive tool (ICING) for visualization and development of models of neurons containing uniquely located ion channels. Here, we illustrate the use of PSICS and ICING, outline the computational strategies used and provide benchmark data for evaluation. We then identify previously unappreciated differences between the effects of stochastic ion channel gating on somatic and dendritic membrane potential activity in several different morphological classes of neuron. We show that the consequences of stochastic gating depend on dendritic morphology and suggest novel functional roles for the kinetics of ion channel gating. Using a previously well-validated realistic model of a CA1 pyramidal neuron we demonstrate that stochastic ion channel gating influences spike output in response to dendritic synaptic input. We show that stochastic gating of axonal or dendritic ion channels substantially modifies synaptically driven dendritic and axonal spike output, with stochastic gating of voltage-dependent sodium and potassium channels having the greatest impact and hyperpolarization-activated channels the least. By demonstrating that neuronal responses to dendritic synaptic input can be intrinsically probabilistic, these results offer a new and general perspective on synaptic integration by central neurons. Full documentation for PSICS/ICING as well as the software, source code and examples are available from the project website (

To investigate the functional consequences of stochastic ion channel gating for neurons with complex dendritic or axonal morphologies, we first developed new software tools that enable accurate, fast simulation (PSICS) and visualization (ICING) of neuronal models that contain stochastically gating ion channels. The organization and development of the new software tools are described in

In PSICS individual ion channels have unique locations that can be viewed along with the compartmentalization chosen for a particular simulation using additional software called ICING. (A) Screen shot of ICING. (B) Detailed view of the compartmentalization of part of the model neuron in (A). (C) Low magnification 3 dimensional detail of dendritic branches from (B). (D) High magnification 3 dimensional detail of dendritic branches in (C) illustrating the location of individual ion channels.

To specify the membrane conductance we adopted a new approach in which the location of each individual ion channel is first uniquely determined (

Since the presently available tools for visualization and development of neuronal models are aimed primarily at deterministic simulations, we developed a graphical tool (ICING) to allow display and manipulation of neuronal models with complex three-dimensional architectures and many discrete membrane ion channels (

We represent ion channels using Markov models, in which each ion channel may be in one of a number of discrete states with the probability of transition to any other state determined independently of the channel's previous history

To first evaluate the modified tau leap algorithm for stochastic simulations we consider a simple three-state Na^{+} channel model recorded with an ideal voltage-clamp (^{+} channels is constant, whereas the equivalent stochastic simulation reveals large fluctuations in the Na^{+} channel current (^{4} simulations were required for the fits to reliably converge to within 1% of the actual values, highlighting the importance of obtaining large numbers of repeated observations for estimating single channel properties using variance-mean analysis. Estimates obtained from 10^{4} or fewer simulations varied around the actual values depending on the number of simulations used (

(A) Examples of simulated stochastic (black traces) and deterministic (red traces) currents in a membrane patch containing 50 stochastic Na^{+} channels with single channel conductance of 20 pS. The membrane potential is clamped at −20 mV. The expanded trace (right) shows the first 5 ms of the compressed trace (left). (B–C) Cumulative estimate of the mean (B) and variance (C) of stochastic currents measured as in (A) are plotted as a function of time. Examples from 5 separate simulations of duration 100 s are shown. Values between 100 and 1000 s are from concatenation of separate 100 s simulations. (D) Examples of 10 simulated current responses (black traces, lower plot), of the membrane patch simulated in (A–C), to a step change in membrane potential from −80 mV to +30 mV (upper plot). The mean (red trace) and variance (blue trace) are calculated from 1000 stochastic current responses. (E–F) Plot of membrane the membrane current variance as a function of the mean membrane current for the rapid activation phase (E) and slower inactivation phase (F) of the 1000 simulated current responses used to obtain the data for (D). The time window for the activation phase is 0–0.3 ms after the onset of the voltage step, whereas the time window for the deactivation phase is 0.3–10 ms after the onset of the voltage step. The number of single channels (N) and the single channel current (I) are estimated from the fit to the simulation data. The red parabola is the variance-mean relationship predicted from I and N of the model and the blue parabola is the fit to the simulation data. (G–H) Estimates for the number of channels (G) and single channel current (H), obtained by variance mean analysis of the inactivation phase of the current responses analyzed as in (D), plotted as a function of the number of simulated responses used for the analysis. Each dot corresponds to a set of data used for analysis. The continuous lines show convergence of the estimates as additional simulations are analyzed up to a maximum of 10^{4} simulated responses. (I) The RMS error, calculated from the difference between the variance mean fit and the expected variance mean relationship, is plotted as a function of the number of simulated responses. The solid lines indicate progressive convergence up to a maximum of 10^{4} simulated responses. For all examples the simulation time step was 10 µS.

Before comparing simulations of neurons with different morphologies, we first established the accuracy of simulation of current and voltage propagation using standard compartmental models for which there are analytical descriptions of the equivalent cable structures ^{+} and K^{+} channels (

(A) Simulated uniform cable with recording and current injection sites. (B) Example membrane potential traces (left) from simulations using leak Na^{+} and K^{+} channels with the indicated single channel conductances. The membrane potential variance is plotted as a function of the single channel conductance (right). (C) Voltage responses to injection of a current step at one end of the cable. Simulated stochastic (Stoch) and deterministic (Det) responses are plotted along with the analytical solution (Ref). Insets show the voltage as the current responses approach their steady-state values. In each case the traces overlap. As a result the Ref and Det traces are obscured by the Stoch trace. (D) Simulated branched cable structure with recording and current injection sites. (E) Voltage responses to injection of a current step at the base of the tree. Insets show the voltage as the current responses approach their steady-state values. Labels are as in (C). Ref and Det traces are obscured by the Stoch trace. (F) Action potentials generated when Hodgkin-Huxley channels are inserted into the cable in A. The top panel compares results of deterministic simulations using NEURON or PSICS, with stochastic PSICS simulations using a single channel conductance of 0.01 pS. The lower panel shows the output of several stochastic PSICS simulations using a single channel conductance of 20 pS. Membrane potentials in (E) and (F) are labeled as in (C), except that the blue trace is data from a simulation using NEURON.

We next assessed simulation of excitable neurons. In a model of a cylinder containing active membrane conductances

While open probability and single channel conductance influence the amplitude of current fluctuations generated by stochastic ion channel gating, little attention has been given to the functional impact of channel kinetics or of interactions between channel properties and the membrane capacitance. The simplified models we used to evaluate PSICS also allowed us to begin to investigate these issues. To avoid non-linearities from voltage-dependent gating, we simulated single-compartment models that contain only passive leak Na^{+} and K^{+} channels. Each channel has one open and one closed state, with an open probability of 0.7, and the relative density of the channels was adjusted to produce a resting membrane potential of −60 mV. We compared a version of the model in which the forward and reverse rate constants for entering the open state were 0.07 ms^{−1} and 0.03 ms^{−1} (slow gating) with a version in which the corresponding rate constants were 7 ms^{−1} and 3 ms^{−1} (fast gating). The model containing the slow gating channels produced membrane currents in voltage-clamp, or membrane potentials in current-clamp, that fluctuated at frequencies below approximately 15 Hz (

(A–B) Examples of membrane currents (top), membrane potential (middle) and corresponding power spectra from 100 s of simulated activity (bottom), for models containing passive leak Na^{+} and K^{+} channels with slow (A) or fast (B) kinetics. The power spectra are shown for simulations with time steps of 10 µs (solid traces), 100 µs and 1000 µs (light traces). The voltage-clamp simulations are of a single isopotential compartment containing 201 Na^{+} and 1407 K^{+} leak channels. The current-clamp simulations are for a cable of length 1000 µm and radius 1 µm, containing 8050 channels distributed across 237 compartments. (C) The error in the mean and variance of a simulated current, mediated by 50 Na^{+} channels clamped at −20 mV for 100 s of simulated time, is plotted as a function of the simulation time step. (D) Power spectra for the currents in (C). Long time steps fail to simulate high frequency current fluctuations and introduce aliasing effects at low frequencies. (E–F) The computation time per simulation time step, required by NEURON (closed symbols) or PSICS (open symbols) for simulations as in (C–D), is plotted as a function of the duration of the simulation time step (E), or as a function of the number of simulated channels when the time step is 20 µs (F). (G) The computation time per simulation time step, required by NEURON or PSICS for simulations as in

To examine how the choice of a suitable simulation time step is constrained by these properties, we initially used the simple passive models described above. For the model containing slow leak channels, simulations with time-steps as large as 1 ms reproduced the dominant components of the power spectra of current and voltage fluctuations (

Since simulation of complex neuronal morphologies can take considerable time, even using optimized computational algorithms, before simulating neuronal morphologies we first investigated strategies to minimize the time required for simulations without affecting accuracy of the results. We evaluated a stochastic implementation of the Hodgkin-Huxley Na^{+} channel model in a single compartment voltage-clamped at a fixed potential (

To determine if improvements in simulation efficiency expected from use of the modified tau leap algorithm and an optimized computational core translate into practical reductions in simulation time, we compared the time required for simulations using PSICS to simulations run in the widely used NEURON simulation environment

Does morphology influence the functional consequences of stochastic ion channel gating? To address this possibility, we first compared the membrane potential noise resulting from stochastic ion channel gating in a hypothetical dendritic tree that obeys Rall's 2/3 power law, with membrane potential noise resulting from stochastic ion channel gating in the corresponding equivalent cable structure ^{+} and Na^{+} channels causes noisy fluctuations in the membrane potential. These fluctuations increase in amplitude by more than ten fold between the proximal and the distal ends of the branched dendrite model (

(A–F) Recordings of resting membrane potential at proximal (grey traces) and distal (black traces) locations on a multi-compartmental model of a cylinder of length 320 µm and diameter 16 µm (A–C) or a hypothetical branched dendrite (D–F). The cylinder in (A–C) is electrically ‘equivalent’ to the dendrite in (D–F), which has a branching organization that follows Rall's 3/2 power law. The distal recordings are from location ‘10’ and the proximal recordings are from location ‘0’. In each panel the membrane potential when the leak channels have fast kinetics (upper traces) is compared to the membrane potential when the leak channels have slower kinetics (lower traces). Membrane potential when the models have a membrane time constant on the order of 0.1 ms (B,E) is compared to models with a membrane time constant on the order of 10 ms (C,F). The scale bars apply to all traces. (G–H) The standard deviation of the resting membrane potential of the models in (A–F) is plotted as a function of recording location. Each point is the average of data from 5 simulations of 1s of neuronal activity. The same data were used for statistical analysis (ANOVA). Black and grey symbols correspond to distal and proximal recording locations as in (A–F) above.

Under what conditions do channel kinetics determine the impact of stochastic ion channel gating? Whereas our earlier simulations indicated an important role for channel kinetics (

Do realistic neuronal morphologies influence the functional impact of stochastic ion channel gating? While the simulations described above suggest this may be the case, they also suggest that the consequences of stochastic gating depend on the specific details of neuronal morphology and ion channel kinetics. To address this question directly, we therefore reasoned that if neuronal morphology is an important determinant of the impact of stochastic ion channel gating, then simulations using identical rules to introduce identical stochastic ion channels into neurons with distinct dendritic morphologies, should predict differences between neurons (

(A) Examples of membrane potential (right) and corresponding morphology (left) from a simulated dentate gyrus granule cell (top), dopaminergic nigral cell (middle) and cortical layer V pyramidal cell (bottom). All models contained identical ion channel distributions. (B) Average membrane potential standard deviation for model neurons of each morphological type plotted as a function of increasing distance along the dendrite from the soma. The membrane potential standard deviation at a particular location corresponds to the right most end of the bar indicated by the corresponding color. The standard deviation increases with distance from the soma.

As the impact of stochastic gating in the abstract models described above depended on channel kinetics (

(A–B) Simulated membrane potential of a model layer V pyramidal neuron (A) and a dentate gyrus granule cell (B) containing either a deterministic leak conductance, fast gating stochastic leak channels or slow gating stochastic leak channels. (C–D) Mean variance of membrane potential fluctuations, recorded from simulated layer V pyramidal neurons (C) and simulated dentate gyrus granule cells (D), plotted as a function of distance from the cell body.

The previous simulations establish that in principle stochastic gating of intrinsic ion channels might be important for neuronal function, but the impact of stochastic ion channel gating on neuronal responses to physiological patterns of synaptic input is not known. We therefore asked if stochastic gating of post-synaptic ion channels in dendritic neurons influence the transformation of synaptic input into spike output obtained with realistic neuronal morphologies and ion channel properties? In the models described so far, ion channel distributions were chosen to facilitate comparisons between morphologies. To address more realistic ion channel distributions we adopt a model of a CA1 pyramidal neuron that has previously been shown to account well for somatic and dendritically initiated action potentials

(A) Morphology of the simulated CA1 pyrmidal neurons (described in

Compared to the deterministic model, the stochastic version generated “extra” spikes at times when the equivalent deterministic neuron was silent and “dropped” spikes at times when the equivalent deterministic neuron fired action potentials (

To evaluate the mechanism for probabilistic initiation of action potentials, we recorded membrane potential from the soma and from proximal parts of each primary dendrite. While some somatic action potentials were preceded by dendritic depolarizations that resemble fully propagating dendritic spikes (

To evaluate the relative roles of stochastic axonal compared with stochastic dendritic ion channels we implemented versions of the model in which one population of ion channels was deterministic and the other stochastic. Both axonal and dendritic stochastic channels caused “dropped” and “extra” dendritic spikes (

(A–B) Rasters for first 20 trials of responses of the CA1 pyramidal neuron simulated as in _{h} are stochastic. Shown are rasters (left), examples of membrane potential responses of the fully deterministic model (red) and first six sweeps recorded from the stochastic model (black), and examples of membrane potential waveforms corresponding to an “extra” action potential triggered by an additional dendritic depolarization. (D) Number of “dropped” and “extra” axonal spikes (top) and dendritic spikes (bottom) during 1s of simulated time for each experimental condition tested. Because of their all or nothing nature, large dendritic depolariations are classified as spikes. ANOVA indicated a significant (p<<1e-9) effect of model configuration on “dropped” and “extra” axonal and dendritic spikes. Key post-hoc comparisons are referred to in the main text. (E) Jitter in the timing of axonal (top) and dendritic (bottom) action potentials for each experimental condition.

Since the model CA1 pyramidal neuron contains several types of ion channel that differ in their kinetics, voltage-dependence and single channel conductance ^{+} channels, followed by A-type and delayed rectifier potassium channels (^{+} channels gated stochastically compared with the fully stochastic model (p = 0.98 and 0.3), whereas there were fewer “extra” somatic and dendritic spikes (p<1e-4 in both cases). Models in which only one of the other ion channel types gated stochastically differed significantly from the fully stochastic model in all measures of “extra” and “dropped” spikes (p<1e-3). Nevertheless, models containing stochastic gating voltage-dependent K^{+} channels generated considerably more than 50% of the number of “extra” and “dropped” spikes observed in the fully stochastic model. Interestingly, stochastic gating of _{h} channels alone had very little impact on axonal spikes or spike jitter, but nevertheless increased the number of “extra” dendritic spikes. The relative lack of effect of _{h} can be explained by the small single channel conductance, while the primary effect on additional dendritic spikes may reflect slow gating kinetics and dendritic localization of these channels (^{+} and K^{+} channels may be particularly important.

To address the functional consequences of stochastic gating of neuronal ion channels we developed and validated new, efficient and general-purpose tools for numerical simulation of cells with complex morphologies. Using these tools we have made several new findings. First, we show that the functional impact of stochastic ion channel gating depends on neuronal morphology and as a result differs between neuronal cell types. Second, we show that depending on a neuron's morphology, ion channel kinetics influence the functional consequences of stochastic ion channel gating. These results suggest that detailed and well-constrained simulations will be important for accurate prediction of the specific functional consequences of stochastic gating in particular cell types. Third, we show that in a realistic model CA1 neuron, stochastic gating of non-synaptic ion channels modifies the timing of synaptically driven somatic and dendritic action potentials, and causes substantial numbers of “extra” and “dropped” somatic and dendritic spikes compared to equivalent deterministic neurons. These results suggests a new perspective on dendritic integration of synaptic inputs, whereby stochastic gating of intrinsic ion channels enables populations of neurons to compute using probabilistic rather than fixed spike codes.

Gating of single ion channels is one of the better-understood stochastic processes in biology

Irrespective of the details of any particular model, our results suggest that neuronal morphology and ion channel properties interact to determine the functional consequences of ion channel gating. Comparison of model neurons with different morphology, but containing identical ion channels, indicates that dendritic morphology plays a key role in determining the functional consequences of stochastic ion channel gating (^{+} channels in _{h} channels gate stochastically, with equivalent models in which other ion channels gate stochastically (_{h} is perhaps not surprising given the relatively small underlying single channel conductance, which is estimated to be on the order of 1 pS _{h} is a major contribution to the resting dendritic membrane conductance of pyramidal neurons

Since it is rarely possible to reduce electrical activity within morphologically complex excitable neurons to tractable analytical models, mechanistic simulation of axons and dendrites relies on well-constrained compartmental models. Compartmental simulations necessarily involve trade-offs between accuracy and simulation time. This is a particularly important problem for simulations that aim to account for the stochastic transitions between the states of each individual ion channel in a realistic model neuron. To enable efficient and accurate simulation, we adopted an algorithm that generates a correct distribution of ion channel states at each simulation time point, while sacrificing explicit representation of ion channel states during the interval between time points. Relatively short time steps are required for accurate simulation of voltage-clamped conductances with rapid kinetics (

Our new approach has several advantages over previous methods for simulation of stochastic ion channel gating. While approaches that add noise terms to the equations used to calculate the membrane currents are computationally efficient

By implementing a previously well-validated model of a CA1 pyramidal neuron using stochastically gating ion channels, our simulation results provide evidence that synaptic integration by dendritic neurons is probabilistic. While the instantaneous output of a single neuron functioning in this way is relatively unreliable, instantaneous representations distributed across a population of stochastic neurons could be read out by summation of their outputs. The impact of such probabilistic integration on information processing and computation by populations of pyramidal neurons remains to be determined. For CA1 pyramidal neurons in the hippocampus, one possibility is that this probabilistic integration is important for encoding of location by the timing of action potentials relative to ongoing network rhythms

Challenges for future experimental and theoretical studies include determining the conditions, additional cell types and sub-cellular locations in which stochastic gating of ion channels affects spike output, and to establish the consequences for computations carried out by neural circuits. At some sub-cellular locations noise introduced by stochastic gating of single ion channels might impose physical constraints on the computational properties of neurons

All calculations were performed with PSICS (

Channel positions are allocated according to user-specified probability densities over the structure such that each channel has a position in three dimensions. The input morphology is sub-sampled at 1 µm for computing local number densities. Axial positions for channels are generated either by sampling a Poisson distribution for the distance to the next channel, or by taking uniform increments to give the desired average density. The angles at which channels occur around a section are allocated randomly from a uniform distribution. At present, these angles only affect the visualization since the structure is later discretized into elements with end faces perpendicular to the axis. The seeds used for the random number generator are stored with the model so that exact allocations can be reproduced. So that allocations are portable across platforms, the generator used is a Mersenne Twister

For computing the propagation of membrane potential changes, the structure is compartmentalized into elements such that all elements have approximately the same value of:

where

Ion channels are represented by kinetic schemes. Each scheme has one or more complexes, and each complex is an inter-connected set of states with expressions for the transition rates between them. Models using the Hodgkin-Huxley gating particle description are mapped to the corresponding scheme where each gate corresponds to a two-state complex

By gathering up the terms, this can be written in matrix form as a master equation

In general,

As this integral is exact for constant

In calculations, significant efficiency improvements over explicit methods can be achieved by using the fixed step transition matrix,

For a stochastic calculation, the element

That is, the elements

For small populations, the update step is applied to each channel individually. For larger populations, significant computation time can be saved by updating only the channels that have a non-negligible probability of changing state during the step and rescaling the bins accordingly. Thus, if the total number of channels in a given state is

For a statistically reliable result, we require a bound on the number of missed events, here set at 2 in 10^{8}. This requires

In general, however, finding the smallest

The first applicable formula is used. If none of the conditions is met, all the channels are advanced individually. These formulae were arrived at by a combination of experimentation and numerical optimization to find expressions that approximate the form of the exactly computed values of ^{8}.

For ion channels the modified tau leap method removes the major cost of the next transition method

For simulations illustrated in ^{+} and K^{+} channels with open probabilities of 0.7. The following channels were included: fast Na^{+} channels (1/µm^{2}); non-inactivating K^{+} channels (0.05/µm^{2}); high-voltage Ca^{2+} channels (0.15/µm^{2}); Na^{+} and K^{+} leak channels (0.016/µm^{2}). The resting membrane potential was set by modifying the ratio of Na^{+} to K^{+} leak channels. In all simulations reported here this was −60 mV. We chose single-channel conductances of 20 pS for all ion channels, as this is similar to values reported for single channel recordings made from neuronal dendrites ^{2} and axial resistivity to 150 Ω cm

The simulations of a detailed model of CA1 pyramidal neuron (_{h} conductance and channel distribution taken from a different publication from the same group ^{+} and K^{+} leak channels were automatically adjusted to achieve a resting potential of approximately −70 mV throughout the cell, while maintaining a total leak conductance consistent with the original model. The single channel conductance of the delayed rectifier K^{+} channels and voltage-dependent Na^{+} channels were set to 20 pS, which is similar to estimates from single channel recordings ^{+} channels and leak channels were also set to 20 pS, which is similar to experimental measurements for D-type channels ^{+} and delayed rectifier K^{+} channels can be considered as fully constrained predictions given currently available data, while our simulations of the fully stochastic model likely estimate a lower limit for the consequences of stochastic ion channel gating. This is because our results from simulations when A/D type K^{+} channels are deterministic, but voltage-dependent Na^{+} or delayed rectifier K^{+} channels are stochastic, nevertheless demonstrate highly probabilistic spike firing, indicating that a smaller single channel conductance for A/D type K^{+} channels would have little impact on the results, while a possible larger single channel conductance for the leak channels would be expected to increase the impact of stochastic gating. Our simulations of A/D type stochastic gating alone should be considered as setting an upper limit for stochastic effects based on known properties of these channels, whereas the simulations of leak channels alone are less well constrained and serve as an illustrative example. Unlike other ion channels, the single channel conductance of _{h} channels is set at 1 pS, which is consistent with noise-analysis of dendritic _{h} recorded from cortical neurons _{h} obtained with cell-attached recordings from CA1 pyramidal neurons ^{2} (1502 in total). Each synapse was activated independently according to a Poisson process with a mean frequency of 5.5 Hz. For analysis dendritic spike times were calculated as upward voltage crossings above a −60 mV threshold. Visual inspection of traces confirmed that this threshold successfully isolated all-or-nothing dendritic events.

Additional analysis of simulation data was carried out using IGORpro (Wavemetrics). Statistical analysis used R (

Overview of PSICS. Model specification files are listed on a green background, simulation outputs on a yellow background and new software components on a clear background.

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Sodium channel model. (A) The sodium channel model used to illustrate ion channel simulation with PSICS has a single open state (O1) connected as shown to two closed states (C1 and C2). (B) The transitions between states of the model are governed by forward α and backward β rate constants that vary as a function of membrane potential (upper graph). The time constants (Taux) and steady-state distribution (Xinf) for each transition are plotted as a function of membrane potential (lower graph). (C) Deterministic currents (bottom) generated by the gating scheme in response to step changes in membrane potential from a holding potential of 80 mV (top). Inset shows the activation phase of the currents on an expanded time base.

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All-or-nothing dendritic responses of a fully stochastic CA1 pyramidal neuron model to synaptic stimulation. (A–B) Membrane potentials recordings from the soma (top) and indicated basal dendrite (bottom) from twenty consecutive trials as in

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Synaptically driven spike output is modified by stochastic gating of single types of ion channel. Raster plots as in ^{+} channels (A), delayed rectifier K^{+} channels (B), A/D type K^{+} channels (C) and leak channels (D).

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Stochastic simulation framework; software development; estimate of the number of ion channels in a central neuron.

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We thank Paul Dodson, Mike Shipston and David Wyllie for comments on a previous version of the manuscript.