MODELING VOLTAGE-DEPENDENT CONDUCTANCES

TO INVESTIGATE ION CHANNEL BEHAVIOR

DURING AN ACTION POTENTIAL

Angie Pronger

December 5, 2007

Abstract

Introduction

Cellular behavior is greatly influenced by the conductances of its membrane. In neural signaling, the conductances partially determine when particular ion channels are activated or inactivated and when maximum activation occurs. By analyzing the shape of a particular conductance curve, it is possible to determine which phase of neural signaling is most affected. For example, the steepest part of the voltage-dependent activation curve for calcium conductance pertaining to the B-current occurs at voltages that correspond to the peak voltages of an action potential (Bower & Beeman). Therefore, the calcium channels pertaining to the B-current achieve maximum activation during and immediately after the peak of the action potential. The curve also demonstrates that it is not possible to activate these calcium channels in the absence of an action potential since activation does not occur at the corresponding voltages.

However, identification of channel activity at particular voltages through examination of conductance curves result in misleading conclusions about neural behavior. Although voltage-dependent conductances help to define the activity of ion channels, neural activity is shaped by various factors. The resting membrane potential, inactivation and activation properties, and the concentration of extracellular and intracellular ions all interrelate and jointly affect the ionic current flow through the membrane. Moreover, these factors show great variation between different species as well as between different kinds of cells within an organism. In glomus cells of the rabbit carotid body, the resting membrane potential is around -48 mv (Overholt & Ficker, 2000), in epicardial cells of the left ventricle, the potential is around -81 mV (Fedida & Giles, 1991). Additionally, neural signaling is highly dependent on chemical and electrical gradients. Firing can also be influenced by ion channel density and the presence and conformation of gates. In the Shaker potassium channel, a change in tilt of the S4 domain is suggested to control the gate of a separate pore domain or remove an arginine residue from the voltage-sensitive domain to allow the influx of ions through the channel (Tambola et al, 2007). The Hodgkin-Huxley model of the action potential incorporates these factors in the four characteristics of action potentials: threshold, all-or-none response, undershoot and the refractory period. The Hodgkin-Huxley equations are as follows (Nelson & Rinzel, 1994):

(i)(a) I= CM dV/dt + gkn4 (Vm-Vk) + gNam3h(Vm-VNa) + gl(Vm-Vl)

______

capacitive current K+ current Na+ current leak current

(primarily Cl-)

Or stated otherwise:

(b) dV/dt = 1/C [Injection current - ∑gi(V) (Vi-V)]

Time and voltage dependence:

(ii) dn/dt = an (1-n) – bnn  gK+ activation

(iii) dm/dt = am (1-m) – bmm  gNa+ activation

(iv) dh/dt = ah (1-h) – bhh  gNa+ inactivation

(n, m, and h are gating parameters)

Thus, the activity of each channel can be interpreted by its voltage dependent conductance of the form gi(v) = gonk, where n (the probability of channels opening), k (the number of gates per channel or cooperative index) and go are changeable parameters. The aim of this project is to assess how the dynamics of each type of ion channel on neuronal firing by creating an electrical circuit that models neural activity.

Design

The electrical equivalent of the Hodgkin-Huxley model is based on the principle that when current flows across the membrane, some of it is used to charge the membrane capacitance and some of it is used to carry ions across. the membrane. Thus, the ionic current is divided into three components: a sodium, potassium, and small leak current that consists primarily of chloride ions. The circuit proposed by Maeda and Makino was modified in the present study and a spike frequency adaptation component was added. (Figure 1).

To best explain what is happening at the electrical level and how this models physiological phenomenon, it is helpful to divide the diagram into three main parts. The capacitor labeled Cm models the membrane capacitance. The greater this value, the longer the circuit takes to charge to reach the voltage provided by the battery the threshold of firing determined by the transistors in the electrical circuit modeling fast Na+ channel. Rleak models the inward leak current. The greater the resistance, the greater smaller the magnitude of the outward flow of negative chloride ions is. The fast inward current of sodium ions is modeled in the first portion of the circuit by transistors BJT1, Digi-Key part no 2N3904, and BJT2, Digi-Key part no 2N3906, and resistors RNa1, RNa2, and RNa3. Modification of these components will determine the shape of the of the depolarization voltage phase of the action potential. The delayed outward potassium current is modeled in the second portion of the circuit by transistor BJT3, Digi-Key part no 2N3904, resistors RK1, RK2, and capacitor CK1. Modification of these components will determine the shape of the action potential during repolarization. The spike frequency adaptation circuit was added to change the firing pattern of the action potential over time.

Properties


A. / Figure 1: Electrical circuit modeling the spike-adapting neuron. A. Circuit schematic. The three ionic channels are highlighted by the red rectangles. From left to right: fast sodium current, delayed potassium current, spike-frequency-adaptation circuit. B. Photograph of the circuit implementation. Corresponding parts of the circuit were also highlighted: blue - membrane capacitor and leakage resistor, red - fast Na+ current, dark cyan - delayed K+ current, light cyan - spike frequency adaptation circuit, lack - voltage divider that allows to linearly change the supply voltage between 0 and 12 V. .

B.

Figure 2 placeholder: a single action potential zoomed-in, side-by-side with a biological action potential - you can steal a pic of that off the web, or from Jimmy-and-Jeff's paper, attached.

Resistors

A resistor is an electrical component that resists current by creating a voltage drop between its terminals. In a voltage divider, two resistors are placed in series with a movable pin in between. As the pin moves up or down, one resistor becomes longer and the other becomes shorter. The longer the wire is, the greater the resistance. Once the pin is adjusted, the voltage divider creates a stable output voltage which is proportional to the input voltage. Furthermore, voltage dividers can create precise reference voltages which are commonly used as inputs to devices of high impedance, so as not to put a high load on the divider.

Capacitors

A capacitor is an electrical component that consists of two parallel plates, each capable of accruing charge. When a voltage source is provided, electrical charges of equal magnitude and opposite polarity build up on each plate and the capacitor begins to store energy. The capacitor will then continue to charge until it has reached the source voltage or until there is somewhere else for the current to go. The voltage across the capacitor is expressed by: during charging and , where is the time constant, equal to the circuits resistance times the capacitance and is the electromotive force, the maximum voltage value of the capacitor.

Transistors

Current within a transistor can be controlled by changing the energy barrier between the p- and n-enriched regions. The energy barrier for the current between collector and emitter is determined by the control voltage on the base. In a neuron, the state of a given channel depends on its previous state of activation. The rate at which it changes between states is dependent upon the energy difference between the states. Therefore, transistors can mimic neural behavior by establishing a threshold under which current will not pass (Mead, 1989).

Figure 3. Input-output characteristic of a bipolar junction transistor 2N3904. The output voltage Vce (collector-emitter), measured through a load resistor ??, rises exponentially once the input voltage Vg (gate voltage) exceeds the threshold (in this case 0.5 V).

Operational Amplifier - LOOK UP IN WIKIPEDIA

-low impedance

Compares the reference voltage to input of neuron, -> fires

Principles of Operation

Current flows from a point of higher voltage to a point of lower voltage. When the 6V power source is turned on, the positive voltage will flow toward the top of the circuit, the inside membrane, through RNa3. Because the membrane voltage at this point is less than the threshold of the diode, current will flow only through Cmem and Rleak. Rleak provides resistance to model leak current of neurons, particularly of chloride ions. Cmem models the membrane capacitance of a neuron. As the capacitor charges, the voltage across the membrane and ground, which models extracellular space, begins to rise. Eventually the membrane voltage becomes great enough to overcome the threshold before the diode. Current will then begin to flow through the diode to BJT1. Once the threshold of BJT1 is reached, current can flow to BJT2. When the threshold for BJT2 is overcome, the opening of its gate will permit current to flow from the battery, through RNa2 and through BJT2 toward the membrane. The opening of this gate creates a positive feedback loop, similar to the one in opening of Na+ voltage-dependent ion channels in the upshoot phase of the action potential (Figure 2). The higher is the voltage differential between the Cmem and the ground, the more the BJT2 gate opens, and the higher is the current that flows through RNa2, thus further increasing V(Cmem). Eventually this voltage becomes high enough to open the gate on BJT3 and activate the delayed K+ current that terminates the action potential (downshoot, Figure 2).

In-Circuit Experiments

First the neural model implemented by Maeda and Makino (Figure 1 minus the spike frequency adaptation circuit) was constructed. An input voltage of 6.0 volts (AC) was used. In order to investigate the behavior of the circuit various parameters were changed: resistors, capacitors, and number of diodes used. For each adjusted parameter, the firing frequency (Hz) was measured, as indicated by the oscilloscope, and the action potential width (sec), the total amplitude of the peak (mV) and total period (s) were assessed.

Additional diodes whose resistances were recorded were then inserted in series before the diode of the fast inward current. In order to obtain a more drastic increase in resistance at the point of that diode, resistors were then used instead of diodes. The effect on the threshold (mV) was observed. Finally, the components of the delayed outward current were modified and the firing frequency (Hz) and action potential amplitude (mV) were recorded.

Results

A. / B.
Figure 4: Effects of the circuit parameter values on the action potential. A. As the resistance of RNa1 increases, the amplitude of the spike increases and reaches a maximum around 2250mV. B. In response to an increasing resistance, the firing frequency is greatly reduced then begins to taper. Correspondingly, the period of each action potential increases. This ranges from about 8.5 seconds at 56kOhm to 20 seconds at 438 kOhm (data not shown).
Figure 5:

While the charge-up phase of the action potential was unaffected by an increasing input voltage, the spike amplitude increased greatly.

Blue squares:

Pink squares:

Correspondingly, the increasing input voltage results in a decreasing action potential width and period.

Figure 6.

By increasing the capacitance of KC, the firing frequency decreases due to its ability to hold a greater amount of charge.

Figure 7. Spike frequency adaptation. Panel A: Voltage across Csfa (spike frequency adaptation capacitor) rises with every spikes as the neuron fires. Once the opening voltage of the transistor BJT4 is reached (marked by arrow), it begins to drain current from the membrane capacitor, thus slowing down the firing (panel B).

Discussion

Spike frequency adaptation is commonly exhibited in neurons of humans as well as other species. ?? >check references, elaborate here<

Discuss firing patterns, i.e. “Action-Potential Broadening and Endogenously Sustained Bursting Are Substrates of Command Ability in a Feeding Neuron of Pleurobranchaea”

*how transferable to robots

*further application (combing project)

*spike-frequency adaptation
*postinhibitory rebound ?
Sources: Jones2001, Angstadt2005.pdf, data and model Sekirnjak2002.pdf , some overview of pacemaking, not sure how useful Saint1998.pdf are modeled using ion channels. model them in hhsim.

Acknowledgments

Appendices

Works Cited

HHSim:

SNNAP (simulator for neural networks and action potentials):

HHSim online w/ equations:

Ion Channels in Bursting Neurons:

Bower, J. & Beeman, D.

Ch. 7 Ion Channels in Bursting Neurons

Digital Neurons (Analog and Digital Neural Models)

VLSI (pdf):

Vertebrate values:

Ion Channels in Human Axon:

Overholt, J., Ficker, E., Yang, T., Shams, H., Bright, G., Prabhakar N. “HERG-Like potassium current regulates the resting membrane potential in glomus cells of the rabbit carotid body.” 2000. Journal of Neurophysiology, 1150-7.

Fedida, D. & Giles, W. “Regional variations in action potentials and transient outward current in myocytes isolated from rabbit left ventricle.” 1991. Journal of Physiology, 191-209.

Tambola, F., Pathak, M., Gorostiza, P., Isacoff, E. “The twisted ion-permeation pathway of a resting voltage-sensing domain.” 2007. Nature, 546-549.

Nelson, M. & Rinzel, J. “The Hodgkin-Huxley Model.” “The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural SImulation System.” Bower, J. & Beeman, D. Sringer-Verlag, New York, 29-49.

Mead, C. “Analog VLSI and Neural Systems.” Addison-Wesley. 1989.

Neuroscience book

Brain and Neuron book

NOTES

Put in graphs, captions

Discussion- SFA

Tomorrow: principles of operation, discussion: robots, further application