Balaton Limnological Research Institute, Hungarian Academy of Sciences, PO Box 35, Tihany, H-8237 Hungary.

Department of Biology, University of York, PO Box 373, York, YO1 5YW, UK.

Abstract

Background

Although octopamine has long been known to have major roles as both transmitter and modulator in arthropods, it has only recently been shown to be functionally important in molluscs, playing a role as a neurotransmitter in the feeding network of the snail

Results

The excitability of the B1 and B4 motoneurons in the buccal ganglia to depolarising current clamp pulses is significantly (P << 0.05) increased by (10 μM) octopamine, whereas the B2 motoneuron becomes significantly less excitable. The ionic currents evoked by voltage steps were recorded using 2-electrode voltage clamp. The outward current of B1, B2 and B4 motoneurons had two components, a transient _{A }current and a sustained _{K }delayed-rectifier current, but neither was modulated by octopamine in any of these three buccal neurons. The fast inward current was eliminated in sodium – free saline and so is likely to be carried by sodium ions. 10 μM octopamine enhanced this current by 33 and 45% in the B1 and B4 motoneurons respectively (P << 0.05), but a small reduction was seen in the B2 neuron. A Hodgkin-Huxley style simulation of the B1 motoneuron confirms that a 33% increase in the fast inward current by octopamine increases the excitability markedly.

Conclusion

We conclude that octopamine is also a neuromodulator in snails, changing the excitability of the buccal neurons. This is supported by the close relationship from the voltage clamp data, through the quantitative simulation, to the action potential threshold, changing the properties of neurons in a rhythmic network. The increase in inward sodium current provides an explanation for the polycyclic modulation of the feeding system by the octopamine-containing interneurons, making feeding easier to initiate and making the feeding bursts more intense.

Background

Molluscan feeding, with its repetitive protraction and retraction of the radula to ingest food, has provided a simple model system for the study of central pattern generator neuronal networks and of the way in which the pattern is reshaped by neuromodulators

The network effects of the OC neurons can be accounted for by two cellular processes. First, they produce an increase in the excitability of the protraction phase interneurons (SO, N1L), so that the same current pulse gives more action potentials. Secondly, they increase the strength of the synapses made between protraction phase neurons, including the synapses from the SO and N1L to the N1M CPG interneuron and B1 motoneuron. Both these effects are mimicked by bath-applied octopamine in the range 1–10 μM, which is below the threshold at which octopamine directly depolarises the membrane potential

An increase in membrane excitability has been found to underlie many behavioural processes, including contributions to the modulation seen during learning

Results

Octopamine increases the excitability of the B1 and B4, but not B2 motoneuron

Excitability experiments were performed in Hi-Di to decrease spontaneous synaptic inputs on the cell under study. This made the neurons silent and only the intracellular pulses were able to evoke action potentials. In Hi-Di saline, a series of depolarising pulses of increasing amplitude was used to determine the threshold and excitability of the B1, B2 and B4 motoneurons (Fig.

Octopamine (OA, 10 μM) modulates the excitability of the feeding buccal neurons

Octopamine (OA, 10 μM) modulates the excitability of the feeding buccal neurons. In the B1 and B4 motoneurons the threshold for action potential generation decreases (A, C), while the B2 motoneuron is less excitable with octopamine in the bath. For the B1 motoneuron the increase of the excitability with octopamine is particularly visible around threshold, while in B4 motoneurons increase of excitability is clear throughout the whole range of the injected current. All control experiments were in Hi-Di saline (to which octopamine was added), and a second electrode was used to inject the constant current pulses to evoke bursts of action potentials. Summary graphs represent dose-response relationship: injected current values (X) versus the mean (± SE) number of action potentials (Y) per 0.5 s depolarizing pulse.

In 10 μM octopamine, the threshold of the B1 and B4 motoneuron is reduced while the B2 threshold is increased. For supra-threshold currents, the B1 and B4 motoneuron generate more action potentials in octopamine than control saline, while B2 generates less. With two to four times threshold stimulus current, these differences are significant (comparison of individual pairs of experimental recordings by the Wilcoxon test: B1 P <= 0.008; B2 P <= 0.008; B4 P < 0.002). With high stimulus currents, the sigmoidal curves for control and octopamine treatments start to merge together. The effect of octopamine on excitability is at least partially reversed by washing for 10 minutes.

Outward currents are not affected by octopamine

When any of the three large buccal neurons, B1, B2 or B4 are held in voltage clamp at -40 mV in normal saline, depolarizing pulses elicit an outward current (Fig.

A series of voltage-activated current traces of B1, B2, B4 motoneurons evoked by 5 mV depolarizing voltage steps between -60 mV and +40 mV

A series of voltage-activated current traces of B1, B2, B4 motoneurons evoked by 5 mV depolarizing voltage steps between -60 mV and +40 mV. A. From a holding potential of -40 mV, initial fast inward current (seen on B2 and B4) is followed by a slowly activating outward current (B1, B2, B4 cells) which does not inactivate during the 50 ms current step. B. The same voltage protocol starting from -80 mV holding potential evokes an additional, fast outward current component on all three motoneurons. C. Subtracting the two current records shows the initial transient outward current is a fast inactivating outward current during the course of the 50 ms voltage steps. All experiments in normal saline.

The sustained outward current activates at about -30 mV, and increases as the step becomes more positive (Fig. _{KV }seen in other molluscan neurons

The outward components of the voltage evoked currents are not affected by octopamine

The outward components of the voltage evoked currents are not affected by octopamine. A. Current-voltage relationship of the peak values of the delayed outward currents recorded on B1 (

Reversal of the outward current in the B1 motoneuron

Reversal of the outward current in the B1 motoneuron. A. Following a 400 ms step to +15 mV, activating the outward current, the voltage was stepped down and the amplitude of the tail current measured immediately upon settling. Upper trace: voltage clamp output, lower trace current. B. The tail current is approximately linear with voltage between -80 and -40 mV, with some rectification, and reverses at -70 mV. In 4 replicate preparations, the mean reversal potential was -67 ± 2.6 mV.

Octopamine selectively enhances the pharmacologically separated fast inward current on B4 neuron

Octopamine selectively enhances the pharmacologically separated fast inward current on B4 neuron. Ai. In the saline containing 50 mM tetraethylammonium (TEA) and 50 μM cadmium, the maintained outward current is reduced and the trace is dominated by a fast inward and transient outward current during five replicate depolarising voltage steps from -80 to -10 mV. (The small inward currents are caused by incomplete voltage clamp because of the high spontaneous activity of electrically coupled B4 cluster neurons). Aii. After 10 μM octopamine is added to this saline the fast inward component is increased without changing the outward component. Bi. In medium containing 50 mM TEA, 4 mM 4-aminopyridine (4-AP) and 50 μM cadmium,, and stepping from -50 to -20 mV, the outward components disappear leaving the fast inward component current (10 replicates). Bii. After adding 10 μM octopamine to the saline, the fast inward current is increased, and a smaller increase is seen in a sustained inward current.

When the B1, B2 and B4 neurons are stepped from a more negative holding potential, -80 mV, the outward current shows an initial peak that is not seen at -40 mV (Fig. _{A }currents first recorded in molluscan neurons by Connor & Stevens _{A }current is well fitted with the product of a single exponential, raised to the power 4 (rising), and a single exponential (decaying). A linear approximation to the time constant gave a good fit over the range -30 to +30 mV. For the inactivation of _{A}, we did not use the _{A}, whereas we found that holding at -80 mV gave a substantial _{A }(Fig _{A }in the B1 motoneuron, giving a half-inactivation value of -71 mV (Fig.

Inactivation of the transient outward current in the B1 motoneuron

Inactivation of the transient outward current in the B1 motoneuron. A. Holding the voltage for 800 ms at different values, and stepping to a fixed potential of +15 mV, determines the inactivation of the transient outward current. The transient outward current was measured at its peak value. B. Summary data for 11 runs from 4 preparations, showing mean ± SE total outward current. The solid line shows a fit of the Boltzmann equation

to the mean data, with a half voltage (

The summary current-voltage relationship (Fig. _{A }to _{KV }(_{A }/ _{KV}), measured from recordings of steps to 0 mV, is larger (1.7) in B2 than in B1 or B4 (both 1.4).

Application of octopamine has no significant effect on the amplitude or activation voltage of either the delayed rectifier or transient components of the outward current in any of the B1, B2 or B4 neurons (Fig.

Fast Inward current increased by octopamine in B1 and B4 but not B2 neurons

From a holding potential of -50 mV, the voltage clamp traces show clear transient inward currents in normal saline in each of the B1, B2 and B4 neurons (Fig. ^{+ }current like those in other molluscan preparations

Analysis of the inward currents of buccal neurons

Analysis of the inward currents of buccal neurons. Ai. A series of voltage-steps from a holding potential of -80 mV to between -35 mV and +5 mV on B1 motoneuron. In normal saline an initial fast inward current is followed by outward current. Aii. In saline containing 50 mM tetraethylammonium (TEA) and 4 mM 4-aminopyridine (4-AP) the inward current component is slightly increased while both (the transient and delayed) outward current strongly inhibited. Aiii Changing to Na free saline with the same TEA and 4-AP concentrations shows only a small transient outward current while the inward current is blocked completely. B. Addition of octopamine to normal saline enhances the amplitude of the fast inward currents in B1 and B4 neurons but not in the B2 motoneuron. Individual current traces of representative experiments from -50 mV stepped to -20 mV, -10 mV and -20 mV, respectively in normal saline. The fast inward current is followed by a slower outward component (Bi). In the presence of 10 μM octopamine the fast inward components are increased on B1 and B4 neurons without substantial changes of the outward currents, while the inward current on B2 neuron is decreased (Bii). C. I-V characteristics of the voltage-evoked fast inward currents show increased peak values of the B1 (Ci, n = 8) and B4 (Ciii, n = 8) neurons in the presence of 10 μM octopamine. B2 neurons (n = 6) show decreased amplitudes of the fast inward current (Cii). D. The time at which the peak inward current was observed is not affected by 10 μM octopamine in the B1, B2 or B4 neurons (same preparations as C). In B, C and D experiments were done in normal saline (NS filled symbols) or normal saline supplemented with 10 μM octopamine (OA empty symbols). Mean ± SE. Asterisks mark significance levels of 5% (*), 1%(**) and 0.1% (***).

The inward current transient was well fitted by the product of a rising exponential raised to the third power times a decaying exponential. The inward current activates and inactivates more quickly than reported in

In the B2 neuron, the maximum inward current is reached at -10 to -5 mV, and this suggests that the B2 inward current is different from the inward current measured on B1 and B4.

Bath applied octopamine, 10 μM, increases the size of the fast inward current in the B1 and B4 neurons, but no increase is seen in the B2 neuron (Fig.

In contrast, for the B2, the inward current in octopamine is not increased; rather it is less than in the control. At the peak current, seen with steps to -5 mV, the current in octopamine is 73% of the control; over the range -25 to +5 mV the inward current in octopamine is 77% of the control (Fig.

In molluscan neurons, the inward current may be carried not only by sodium but also by calcium ions ^{+ }current (Fig.

Basic physiology of the B1 motoneuron

The resting potential of the B1 motoneuron was 55 ± 1.7 mV (mean ± standard error, SE, N= 16 preparations). Action potentials had a threshold of 1.76 ± 0.23 nA (mean ± SE; N = 15). The mean (± SE) time constant for hyperpolarising pulses was 54 ± 5 ms (N = 11 trials in 7 preparations).

Simulation

In voltage clamp simulations, the membrane potential was stepped from a holding voltage of -80 mV to -30 mV, and the simulation repeated, incrementing the step potential by 5 mV. The calculated sustained and transient outward currents (Fig. _{A }current increases more slowly below 0 mV than the observed data, but becomes larger than the observed current thereafter. Simulations run from -40 mV (instead of -80 mV) show the _{A }reduced from 260 to 3 nA peak for the step to +10 mV. Under voltage clamp, the calculated inward current (Fig.

Simulation of ionic currents under voltage clamp in the B1 motoneuron

Simulation of ionic currents under voltage clamp in the B1 motoneuron. A. Transient currents from a holding voltage of -80 mV, showing a series of steps starting at -30 mV and increasing in 5 mV increments. Ai. Sustained outward current, the sum of two (increasing) components; compare the B1 data from Fig. 2A. Aii. Transient outward current, rapidly activating and then decaying; compare to data from Fig. 2C. Aiii. Inward current, showing more rapid activation and inactivation as the steps increase; compare to B1 data in Fig. 6. B. Total current in simulation (bold dotted line) from -50 mV to -25 mV, -15 mV, -5 mV and +5 mV, compared with current recordings from 2 different preparations (thin solid lines). At -25 mV the real data show the effects of an action potential propagated from the contralateral cell, starting about 1.5 ms into the recording (so only one preparation is shown at this step). The bold line in black shows the zero current level. C. Simulated Current – Voltage plot, showing the steady-state outward current (IK) the inward current (INa) the leak current (ILeak) and total current (ITotal). The steady state value of the transient outward current is too small to show on this scale. The steady state total current crosses the x-axis at -52 mV with a positive slope, giving this as the calculated stable resting potential.

We have compared the calculated voltage clamp response with that from two representative preparations which had a holding potential of -80 mV (Fig.

Solving the current clamp equations for the resting potential gave a stable value of -52.5 mV (Fig. _{K}, and the leak current dominate. The resting conductance of 88 nS, is the same as the value measured in seven preparations using a series of 6–8 hyperpolarising pulses 1 to 3 nA, 1 to 3 s in duration, which gave a mean B1 conductance of 87 ± 12 nS (equivalent to 13.7 MΩ). The capacitance of the cell was set at 3.5 pF (estimate from hyperpolarising pulses: 4.2 ± 0.4 pF).

In current clamp simulations, positive currents led to depolarisation of the cell membrane. Above 1.55 nA action potentials were produced 25 ms in duration, rising to +20 mV, with a noticeable after-hyperpolarisation (Fig.

Simulation of constant current injection into the B1 motoneuron in normal saline and with octopamine

Simulation of constant current injection into the B1 motoneuron in normal saline and with octopamine. A. A current pulse of 1.6 nA with

To mimic the effect of 10 μM octopamine, the simulation was run with the maximal sodium conductance (

In the snail feeding system, prolonged depolarisation may occur through the tonic release of octopamine, serotonin, peptides or other neuromodulators. Although this might be expected to increase the firing of the cell, this is not always the case. In Hodgkin-Huxley simulations, addition of steady depolarising currents can lead to a reduction in activity, because the sodium channels become inactivated. We therefore modelled the effect of a continuous 0.5 nA depolarising current into the B1 on the number of action potentials produced by sharp current pulses. This changes the resting potential to -47 mV; it also lowers the threshold and increases the firing rate (Fig.

We also tested the effects of increasing the sodium channel conductance in two publicly available models of other molluscan neurons. The simplest model is the space-clamped Hodgkin-Huxley model of a squid axon which has only a single inward and single outward current. Running the implementation by Bezanilla

Discussion

Octopamine enhances the excitability of B1 and B4 motoneurons

We have extended our previous observations that low concentrations of octopamine increase the excitability of buccal feeding interneurons

Overview of the voltage activated currents in B1, B2 and B4

The ionic currents in large molluscan neurons have been extensively analysed ever since the 2 microelectrode voltage clamp technique was introduced in the 1960s

Outward currents

Both transient (_{A}) and delayed outward potassium currents (_{KV}) are present in all three buccal motoneurons (B1, B2 and B4). _{A }and _{KV }activate at -45 and -30 mV respectively and are blocked separately by 4-AP and TEA. The transient current is inactivated at a holding potential of -40 mV. The delayed potassium current has two kinetic components, one fast and steeply voltage dependent, and a slower one, which may be due to a calcium-activated potassium current. This would fit with both the kinetics and with the I-V curve from three neurons. In these, the IV curve was extended to +120 mV (data not shown) and it sagged above +80 mV. A 30 ms prepulse to 0 mV abolishes the sag, and this suggests a role for a calcium-activated potassium current

Each cell type has its own typical shape to its outward current trace. The most characteristic difference is that the transient current in the B4 motoneuron inactivates more quickly than that of the B1 or B2 neurons, and this may indicate differences in the _{A }subtypes expressed in the buccal motoneurons _{A }/ _{KV }and this is larger in B2 than B4, implying that expression of the channels are controlled independently.

We have no data suggesting the presence of two other common channel types, either S-channels or inward rectifiers, though Straub & Benjamin _{h }current in B4 from current clamp recordings. Hyperpolarizing steps to -100 to -150 mV showed no activating currents in any of the B1, B2 or B4 motoneurons under our conditions.

Inward currents

An inward transient sodium current (_{Na}), is the main inward current in the B1 and B4 cells, with the peak current seen in steps to -25 to -15 mV. However, the fast inward current in the B2 neuron differs, with the peak current occurring at less negative potentials, (-10 to -5 mV). This suggests that the B2 inward current may be a mixture of _{Na }and a high-voltage activated (HVA) Ca^{++ }current. HVA Ca^{++ }currents have been reported from isolated buccal neurons

Effect of octopamine

At 10 μM, octopamine does not affect the resting membrane potential of the buccal neurons B1 and B4 although this concentration is enough to modulate their excitability. 10 μM octopamine is also enough to produce significant changes in the ionic currents evoked by voltage steps in the B1 and B4 motoneurons.

In the B1 and B4 neurons, low concentrations of octopamine, affect the inward, but not the outward currents. The mean value of the inward Na^{+ }current is increased by 33% (B1) – 45% (B4), and the difference is significant at the 0.5% level for one and two data points. The octopamine I-V curve appears as an amplified version of the control, with similar maximal activation voltage. Our simulations show that the effects produced by low concentrations of octopamine on the voltage activated currents will be synergistic to the effects of high (100–500 μM) concentrations of octopamine on membrane potential, a depolarisation of about 10 mV

In the B2 neuron, 10 μM octopamine has no effect on the voltage-activated outward currents. However, the B2 motoneuron differs from the B1 and B4 neurons, in that low concentrations of octopamine do not increase the fast inward current. In fact, octopamine reduces the fast inward current in the B2 motoneuron, though this is only just significant at the 5% level at two data points.

The effects of high concentrations of octopamine (100 – 500 μM) on the B2 motoneuron are also different to their effect on the B1 and B4 cells

Simulation of B1

We have modelled the voltage-activated currents of B1 motoneuron using a Hodgkin-Huxley style simulation, with a view to confirming that the 33% increase in inward sodium current confers an increase in excitability. In this model, we included a sustained outward current (with two kinetically separate components) and a transient outward current as well as the inward current. The architecture of this model resembles those devised by Connor & Stevens

The main purpose of the simulation is to test in a quantitative manner the effects of increasing the inward sodium current on excitability. The simulation clearly shows that a 33% increase the maximum inward sodium reduces the threshold and enhances the firing rate in response to the same constant current stimulus. This increase in excitability when the inward sodium conductance is increased is preserved in our simulation when it is started with different parameters, for example when the maximum sustained outward current is reduced. When a small depolarising current is added to the simulation, a further increase in excitability is seen.

A decrease in threshold and increase in the spikes elicited by constant current pulses when the inward sodium current is increased is also seen in two other molluscan action potential simulations, of a gastropod sensory neuron and of the squid giant axon. Thus an increase in inward sodium current seems to be a general mechanism for an increase in excitability, as it is seen in all 3 models we have used. Conversely, a 33% reduction in the inward sodium conductance reduces excitability and increases the threshold.

Mechanism(s) for octopamine to increase B1 and B4 excitability

The octopamine – induced increase in the fast inward current means that depolarizing synaptic inputs will open more Na^{+ }channels and so be more likely to generate an action potential in the presence of octopamine. This provides a straightforward explanation of the increase in excitability, which was manifested in B1 and B4 as more action potentials were evoked for the same depolarising stimulus. We have confirmed this effect using in a quantitative simulation. Our observation of octopamine-induced increase in _{Na }in two cell types suggests that a similar cellular mechanism may underlie the increase of excitability of the SO, N1L and N1M. Our simulation also supports the idea that the decrease in excitability in the B2 motoneuron may result in part from the small decrease in inward current.

Comparison with other mechanisms to increase excitability

A very wide range of neuromodulators have been shown to control the excitability of neurons in snails, crustacea and vertebrates. Among these, modulation of inward currents has been demonstrated by amines _{KS }underlies the increase in excitability, but an independent reduction in _{KV }increases spike width – for review see _{A }and the calcium-activated K^{+ }current during conditioning

As well as changes during conditioning, modulation of neuronal excitability plays a fundamental part of the reconfiguration of neuronal networks. Again, in many model systems, control of voltage activated channels occurs in synchrony with effects on resting membrane potential. This has been extensively explored in the stomatogastric ganglia of crustacea

Conclusion

We have shown that low concentrations of octopamine modulate the excitability of motoneurons in the snail feeding system. The changes in the inward sodium current quantitatively account for the increased excitability of the B1 motoneuron. Thus, for the first time in molluscs, we are able to relate voltage clamp data using a quantitative simulation to a rhythmic network where we have a good data on how behavior is changed by the modulatory effect of octopamine

Methods

Snails

Pond snails,

Dissection

Experiments were done at room temperature to which the snails had been acclimatised for > 24 hours. The CNS, including the buccal ganglia was dissected free of other tissue and pinned out in a Sylgard dish through which saline could be pumped. The outer (white spotted) layers of connective tissue were removed from the buccal ganglia with forceps, and the inner layers digested for 2–5 minutes with 0.1% Sigma Protease XIV.

Current clamp

The large buccal neurons were identified visually from previous maps

Voltage clamp

Two-electrode voltage clamp was performed as described for other

Our voltage clamp analysis is restricted by the way that many buccal neurons are electrically coupled. This includes the B4 neurons, which are coupled, not only to their contralateral partner, but also to the B4 cluster neurons that surround them. This has a major impact under voltage clamp, with traces being contaminated with action potentials from the surrounding cells (e.g. Fig

Saline solutions

Under control conditions, standard

Final concentrations (mM) of

**Normal saline**

**Hi-Di saline**

**TEA+ 4-AP**

**Na-free TEA + 4-AP**

**TEA + Cd**

**TEA + 4-AP + Cd**

NaCl

24

24

24

--

24

24

KCl

2

2

2

2

2

2

CaCl_{2}.2H_{2}O

4

14

4

4

4

4

MgCl_{2}.6H_{2}O

2

8

2

2

2

2

NaH_{2}PO_{4}.2H_{2}O

0.1

0.1

0.1

--

0.1

0.1

NaOH

35

35

35

--

35

35

KOH

--

--

--

40

--

--

HEPES

50

50

50

50

50

50

Tetraethyl-ammonium (TEA) chloride

--

--

50

50

50

50

4 amino-pyridine (4-AP)

--

--

4

4

--

4

N-methyl-D glucamine

50

CdCl_{2}

0.05

0.05

Composition of the

Once a stable recording had been reached, 10 μM octopamine was added to the saline being pumped into the bath and the measurements repeated.

All chemicals were from Sigma.

Modelling

A Hodgkin-Huxley simulation of the membrane currents in the B1 motoneuron was implemented in the computer algebra package Maple (version 8)

Hodgkin-Huxley simulations assume that the proportion of channels open (through activation and inactivation by gates) follows the Boltzmann distribution, regulated by a first order ODE, and that the current flowing through each kind of channel is given by Ohm's law (see Appendix 1). We used this set of equations in the voltage clamp worksheet. In the current clamp worksheet, another ODE, the Capacitor Equation, determines the voltage change from the total current and the membrane capacitance, is added. Our worksheets follow formulation of these equations laid out by Connor & Stevens

Our worksheets contain two outward currents (sustained and transient) and an inward current because our Results (see below, Figs.

The parameters for our equations were estimated by extracting the voltage clamp data into Excel (Microsoft, Seattle, USA).

The time constants of the outward currents were estimated by fitting exponential curves (raised to integer powers) to the voltage clamp data using the least squares method and the Solver tool. The Solver tool was also used to fit the Boltzmann models (Equations 1 and 3), to give the voltage dependence of the open probabilities of the channels and time constants. For the outward currents a linear approximation of the Boltzmann equation relating the time constant to membrane potential (Equation 3) was valid over the physiological range of membrane potentials. For the inward sodium current, the time constants of Connor & Stevens

The leak conductance and membrane capacitance was estimated from fitting a single exponential hyperpolarising -1 to -2 nA current clamp stimuli.

The exact equations used and numerical constants are shown in Appendix 2, copied from Maple rtf format output.

We also tested the effects of numerical changes to the inward sodium current on threshold and firing rate in two other publicly available models: the space-clamped squid giant axon

Appendix 1 Outline of Simulation

Hodgkin-Huxley style simulations are based on the idea that ions cross the membrane by flowing though channels controlled by independent gates. The Boltzmann equation gives the steady state proportion of gates (_{inf}), which will be open at a particular transmembrane voltage (

where the parameters

In this equation, the time-constant,

In a neuron, the summed conductance (

^{n }

while for a channel controlled by multiple types of gates (like the inward sodium or transient outward currents)

where

This set of equations, replicated for each kind of channel, is sufficient for a voltage clamp simulation, but to calculate a current clamp response, a further ODE is required, the Capacitor equation, which gives the rate of change of voltage from the capacitance (

where the overall current is the sum of those flowing through each channel (Σ I) less the stimulus current (_{stim}).

These equations were implemented in Maple (typical output in Appendix 2). The Additional Files show worksheets (in mws [Maple worksheet] and mpl [Maple input] formats) and further output (in pdf format).

Appendix 2 Maple Output for simulation of B1

This output shows the exact constants and equations used for the simulation shown in Fig.

Constants

Stimulus current, in nA, at 0.1 ms goes from zero to 1.6 nA

60

Octopamine ligand gated current

Capacitance in microF

Equilibrium voltages, mV

Sodium, Potassium, Leak

28

Fixed leak conductance

Sodium current

maximum conductance, m and h components

with max value and time constant as functions of voltage

Sustained Potassium current

maximum conductance, NA and NB components

with max value and time constant as functions of voltage

Transient Potassium current

maximum conductance, a and b components

with max value and time constant as functions of voltage

Initial conditions, start from equilibrium voltage (-52.5 mV)

Sodium current

Sustained Potassium current

Transient Potassium current

Current equations

Sodium current

Sustained Potassium current

Transient Potassium current

Leak current

Total ionic current

^{3 }h(

^{2 }+ 2.880 NB(

^{4 }b(

^{3 }h(^{2 }+ 2.880 NB(

+12 (v(^{4 }b(

Differential equations

ODEs for Sodium current

ODEs for Sustained Potassium current

ODEs for Transient Potassium current

ODE for Voltage as function of current

Stimulus and octopamine ligand gated currents included here

- 2000.000 (v(^{3 }h(

- 285.714 (v(^{2 }+ 2.880 NB(

- 3428.571 (v(^{4 }b(

Solve system of ODEs

**proc**(**end proc**

Plotting...

Setup plot

"'if(t <. 1,0,1.6)"

"7.0"

Plotting voltage vs time

[

{Fig.

>

Authors' contributions

AV and CJHE devised the study. Most voltage clamp experiments were done by AV, with support from HS. CJHE carried out the simulation, and drafted the manuscript. AV revised the text, and all authors contributed to its final version.

Maple worksheet for the ionic currents in the B1 neuron, including graphs of the time constants and equilibrium proportion of gates open ionic currents against holding voltage.

Click here for file

Maple commands for Additional File 1, for non-GUI interfaces.

Click here for file

Maple output from Additional File 1, captured and converted to PDF.

Click here for file

B1_vc.mws. Maple worksheet for voltage clamp simulation of the B1 neuron, in this case stepping from -50 to -10 mV. Maple users can set v0 and v to the holding and pulse potentials (in mV) and run the simulation. The Maple output includes a current / time plot.

Click here for file

B1_vc.mpl. Maple commands from Additional File 3, for non-GUI interfaces.

Click here for file

B1_time.mws. Maple worksheet for current clamp simulation with the current initially zero and then stepped to 1.6 nA. The output of this worksheet is shown in Appendix 2 and the plot in Fig.

Click here for file

B1_time.mpl. Maple commands from Additional File 3, for non-GUI interfaces.

Click here for file

Acknowledgements

This work was supported by the Wellcome Trust Collaborative Initiative Programme. We are grateful to Prof. Baxter for his help with the SNNAP simulation.