Neurophysics

Neurophysics

Adrian Negrean

doctoral student

CNCR, VU University Amsterdam

Office address:

Department of Integrative Neurophysiology (INP)

Centre for Neurogenomics and Cognitive Research (CNCR)

Neuroscience Campus Amsterdam (NCA)

VU University Amsterdam

De Boelelaan 1087

1081HV Amsterdam

adrian.negrean@cncr.vu.nl

Contents

1.Aim of this class

2.A first order approximation of neuronal biophysics

2.1Introduction

2.2Electro-chemical properties of neurons

2.3Ion channels and the Action Potential

2.4The Hodgkin-Huxley model

2.5The Cable equation

2.6Multi-compartmental models

3.Bibliography

1.Aim of this class

The study of brains and cognitive processes has been traditionally the preoccupation of biologists and psychologists and only recently, in the last 60 years these issues started to be looked at from a physical perspective. There has been a lot of effort in popularizing physics and mathematics in the biology community (such as in the field of computational biology) and many good textbooks have been written with the aim to present various physical and mathematical concepts (dynamical systems, graph theory) to biologists in a friendlier manner. However amusing it may seem, there is also a need to present biology to physicists in a friendlier manner as well, and my task for the next classes will be therefore to convince you that there’s a lot of physics going on in the brain.

2. A first order approximation of neuronal biophysics

2.1 Introduction

Neurons share many features with other cells, such as having a cellular membrane composed of phospholipids that separates the cytosol containing various cellular elements from the environment. A striking difference between neurons and other cells is their characteristic morphology (Fig. 1), with multiple branches (2, 3 and 4) that extend from the cell body (1), connecting neurons together via synapses (5) through which neurons communicate with each other.

In a simple picture, the dendrites of a neuron on which axons of other neurons form synapses, represent the ‘input’ to the neuron that the cell body ‘integrates’ and then ‘outputs’ the result via the axon to other neurons. The problem with this simple picture that will have to suffice for the moment is that the shaping of synaptic inputs is already happening at the level of dendrites and that the whole neuron takes part in ‘processing’ its inputs. This ‘little’ detail is often ignored in the field of artificial intelligence that uses overly simplistic models of single neurons to justify their use in large scale neuronal networks simulations.

(4)

(3)

(5)

(1)

(2)

Figure 1: A histological staining of a single so called pyramidal neuron from a mouse observed under a light microscope (photo credits: Cristiaan de Kock, CNCR, VU, Amsterdam). (1) Cell body, (2) Axon, (3) Apical dendrite, (4) Basal dendrite, (5) Synapses.

To a large extent communication between neurons occurs through the initiation of an Action Potential (AP) in the axon initial segment which is an electro-chemical wave of excitation that propagates throughout the axonal tree much like the ignition of a dynamite fuse. When this wave of excitation reaches a synapse, it causes the release of neurotransmitters that in turn activate electrochemical processes in the post-synaptic neuron. Depending on the type of neurotransmitter, the postsynaptic neuron may be excited or inhibited to produce an AP.

Check out this link for some nice animations and explanations (report if broken): http://www.youtube.com/watch?v=DF04XPBj5uc&feature=related

There’s a great diversity of neurons (Fig. 2) that can be readily observed from differences in morphology, anatomical location and function which can often make their classification difficult. Moreover, at the level of a single neuron, there is a great diversity of cellular components that will determine the input-output transformations of a neuron, i.e. the way it reacts to synaptic inputs. A full understanding of how brains work must be able to explain how these details fit together and at the same time by accounting for all the details, it is often easy to miss the big picture.

Figure 2: Diversity of neurons that can be readily distinguished based on their morphology.

2.2 Electro-chemical properties of neurons

The neuronal membrane like other cellular membranes is composed of a bilayer of phospholipids that insulates the cytosol from the external environment (Fig. 3). The insulation provided by the phospholipid membrane allows the cell’s cytosol to have a different ionic composition than the external medium (i.e. different concentrations of Na+, K+, Ca2+, Cl-). As it will be explained shortly, this ionic imbalance will lead to an electric potential difference across the membrane, which in turn will create an electric field across the membrane. Considering these properties, the cellular membrane will act as the electrical equivalent of a capacitor. The capacitance of a simple cell having a soma with a surface area A can be calculated from the well known formula of parallelplate capacitors:

Where ε is the dielectric constant of the phospholipid membrane, ε0 , the dielectric constant of vacuum and d the thickness of the membrane (which is about 2.3 nm). A handier way of calculating the capacitance of a simple cell is to remember that cellular membranes have a specific capacitance of about 1.0 μF/cm2 which can be multiplied with the surface area of the cell.

Figure 3: The neuronal membrane is composed of a bilayer of phospholipids that insulates the contents of the cytosol from the environment.

The insulation of cellular membranes is not perfect and additionally the membrane contains many pore-forming proteins through which ions can flow that are collectively referred to as ion-channels. Ion-channels can actively control the flow of ions depending on factors like the membrane potential in the case of voltage-gated ion-channels (Fig. 4), external ligands such as neurotransmitters that

bind to ligand-gated ion-channels, intracellular factors in the case of metabotropic ion-channels or simply, their conductance remains constant, referred to as passive ion-channels or leak-channels.

Figure 4: Structure of a voltage-gated potassium selective ion-channel from the Kv1.2 gene. A: Side-view with the extracellular part above and the intracellular part below. B: Top-view from the extracellular side. The channel is a tetramer, each subunit shown in a different colour with TM representing the integral membrane component of the complex, T1 a cytoplasmic domain, a β subunit and transmembrane helices S1-S6 [Long et. al. 2005].

The membrane capacitance together with leak-channels (detailed in the next section) gives rise to the passive electrical properties of the cell (Fig. 5). Note that the electrical circuit shown in (Fig. 5) includes a voltage source Vrest which takes into account that there is a potential difference between the two sides of the membrane.

A B

Figure 5: A: Electrical circuit equivalent of a simple cell with leakchannels. B: Basic electrical circuit describing a small membrane patch.

As it has been mentioned before, the membrane potential is due to a difference in the ionic composition of the cytosol and the external solution. When a salt, such as NaCl is dissolved in water, it dissociates into its ionic constituents Na+ and Cl- and the solution is overall electrically neutral. Let’s make a thought experiment that will clarify the relation between ions and electrochemical potentials: suppose that you have a jar filled with water, which is separated in two compartments by a membrane permeable only to Cl- and you add NaCl only to one of the

compartments (Fig. 6). Initially all the Na+ and Cl- ions will be in one compartment and each compartment will contain the same number of ions, making the two compartments electrically neutral (Fig. 6A). As time passes, Cl- ions will diffuse across the membrane (increasing the entropy of the system) and an electric potential will develop between the two compartments. The increase in potential difference creates in turn an electric field across the membrane that will oppose the increase in ionic imbalance. Eventually the system reaches a dynamic equilibrium, where diffusion of Cl- is balanced by the electric potential difference.

Cl- permeable membrane

Figure 6: Ions and electrochemical potentials. A: When salt is added initially only to one compartment, both compartments are electrically neutral and there is no potential difference between them. B: As time passes, Cl- ions diffuse across the membrane permeable only to Cl- and a potential difference between the two compartments will be established. In the end the system reaches equilibrium where diffusion across the membrane is balanced by the established electric field across the membrane.

The value of the equilibrium potential Eion of a system similar to (Fig. 6), such as a simple cell, having a membrane permeable only to a particular ion can be calculated using Nernst’s equation:

where z is the valence of the ion, R is the gas constant (8.315 J K-1mol-1), T is the temperature (K), F is Faraday’s constant (96.485 C mol-1) and [ion]o and [ion]i are the concentrations of the ion inside and outside of the cell respectively.

The cytosol of a realistic cell and the extracellular medium contain several ion species such as K+, Na+, Cl-, Ca2+ (Fig. 7) and when their membranes are simultaneously permeable to several ions (formula valid for monovalent ions! for divalent ions there a more complicated formula), the steady state equilibrium potential (or the cell’s membrane potential) is described by the Goldman-Hodgkin- Katz (GHK) equation:

where in addition to the terms described in Nernst’s equation, the relative contribution of each ion is determined also by the relative permeability of the membrane. For example, in the squid giant axon, where the propagation of neuronal impulses termed Action Potentials (AP’s) has been first described, at resting membrane potential (no AP) the membrane relative permeability ratios are pK:pNa:pCl=1.00:0.04:0.45. Thus, the cell has two means of controlling its membrane potential: 1) by changing the ionic composition of its cytosol, 2) by changing the membrane permeability to different ions using ion-channels that were introduced previously.

Figure 7: Intracellular and extracellular concentrations of different ions (millimoles) given in parentheses for a typical mammalian neuron and their Nernst equilibrium potentials (mV). The permeability of the membrane to different ions is regulated by various ion-channels inserted in the membrane and the ionic composition of the cytosol is actively maintained through ionic pumps, in this case, a Na+-K+ exchange pump.

To understand better the relation between membrane permeability to different ions, equilibrium potentials and movement of ions, I’ll give an example situation based on a cell having typical ionic concentrations and equilibrium potentials shown in (Fig. 7). Suppose that based on the GHK equation, the calculated membrane potential is -64 mV. This membrane potential is more positive than the equilibrium potential for K+ of -102 mV, which has the consequence that an increase in the membrane permeability for K+ will cause an outward flow of K+. Only if the membrane potential would be forced to be -102 mV, a change in the permeability for K+ would cause no flux of K+ across the membrane. What would happen with K+ if the cell is hyperpolarized to -120 mV?

2.3 Ion-channels

To make an analogy between electrical circuits and neurons, ion-channels are to a neuron what transistors are for microprocessors in computers. Ion-channel function forms the basis of understanding how neurons respond and adapt to synaptic inputs and any theory aiming to describe how brains work, must take this aspect into account.

As briefly introduced before, ion-channels are membrane-bound proteins that have three important properties: 1) they conduct ions, 2) they are selective for specific ions (Fig. 8) and 3) they open and close in response to specific electrical, mechanical or chemical signals.

Figure 8: Ions can cross the plasma membrane only through specialized poreforming proteins called ion-channels. Ions in solution are surrounded by several water molecules forming a hydration shell that prevents the ion to cross the lipid membrane by creating a large potential energy barrier. A binding site inside the pore selects a certain ion based on the size and charge of the ion, hydration shell size and hydration energy.

Early studies of ion-channel properties revealed that the flow of ions can occur in discrete steps depending whether the channel is in an open or closed state. In these experiments (early 1960s), a thin lipid bilayer created over a small hole in a non-conducting barrier between two salt solutions contained a low concentration of a 15-amino acid peptide, gramicidin A, that had the property of forming pores when two-such peptides associated in the membrane (Fig. 9C). Creating a potential difference between the two salt (e.g. NaCl) solutions, the occasional formation of a pore could be observed as a discrete increase in current (Fig. 9A) that lasted a certain amount of time until the two subunits of the pore dissociated again (note that there is a dynamical balance between the

single and associated form of the peptide, depending on the reaction rates). Varying the potential difference across the membrane, the peak current flowing through the channel varied linearly (Fig. 9A,B) and current-voltage relationship could be described by Ohm’s law i=V/R, demonstrating that such an ion-channel behaves effectively like a resistor in an electronic circuit. Now it becomes understandable that a resistor has been included in the electrical circuit equivalent of a simple cell to model leak-ion-channels (Fig. 5B).

Figure 9: C: Gramicidin A peptide has been added to a phospholipid bilayer membrane to form trans-membrane channels that allow passage of ions. A: The formation of functional Gramicidin A channels can be seen as random step-increases in current when a potential difference is applied to the membrane. B: The size of the current steps is related to the applied potential through Ohm’s law.

Gramicidin A is a cation-selective channel (check Wikipedia if you want to know what it’s good for). Looking at the single-channel currents recorded at various potentials in (Fig. 9A), what is the reversal potential for this current? Generally when describing single channel currents, the reversal potential (see eq. (2) and eq. (3)) has to be also taken into account since there may be an ionic imbalance between the two sides of the membrane:

resistance), E the holding potential and Erev the reversal potential of the current.

While leak ion-channels having an Ohmic behaviour are the simplest to describe, the great majority of ion-channels require more elaborate biophysical models to describe them. In the following I will focus on the biophysical modelling of two voltage-gated ion-channels that are essential in the production of action potentials that neurons use to communicate with each other. First I will give an overview of what happens during an action potential and then we’ll have a closer look at what the ion-channels are doing. Don’t worry if it’s not immediately clear, I’ll explain it step by step.

After the activation of several excitatory synapses of a neuron, a depolarizing current travels through the dendritic tree and depolarizes the soma which had a resting membrane potential around