Neurophysics

-64 mV (Fig. 10). When the somatic membrane potential reaches about -45 mV, Na+ permeable channels open, Na+ ions enter the cell and further depolarize the soma such that within a millisecond the membrane potential reaches a value around +20 mV. With a slight delay, K+ permeable ion channels open and the outflow of K+ ions repolarises the membrane potential to a value close to -64 mV ending the action potential.

Let’s start with the biophysics of K+ channels involved in action potential generation. During the opening and closing of ion-channels, several subunits within the membrane-spanning protein must change their conformation (shape) at the same time for the channel to open or close. In the simplest case then, assuming that the subunits change their conformation independently of each other, the probability for the channel to open is:

where k is the number of subunits (in this case k=4) and n is the probability that a single subunit switches to an open conformation. The subunit probability n is related to the membrane potential through a kinetic scheme where the closed to open transition occurs at a voltage-dependent rate αn (V ) and the reverse transition open to closed occurs at a voltage-dependent rate βn (V ) .

In practice, the rate coefficients αn (V ) and βn (V ) are obtained from fitting experimental data,

however thermodynamical arguments can be also used to a good approximation to describe the rate coefficients. Voltage-gated ion-channels are able to respond to a change in the membrane potential by having one or more charged domains (charged amino acids) that couple with the membrane electric field. Thus, a conformational change involves the movement of a certain fraction Bα of

charge q within the membrane electric field. The potential energy difference (barrier) qBαV that

separates the two conformational states is then proportional to the membrane potential. This means that the probability of thermal fluctuations to have enough energy to overcome this potential barrier and change the conformational state of the subunit, will be proportional to the Boltzmann factor exp(−qBαV / kBT ) . Using such a thermodynamical argument,αn is expected to have the form:

You may have already come across a similar equation in chemistry where it is known as the Arrhenius equation describing reaction rates. A similar equation can be written for βn (V ) .

A B

(1)

(2)

(3)

(4)

time

(5)

Figure 10: The production of action potentials in neurons. A: (1) Incoming action potentials release an excitatory neurotransmitter (for example, glutamate) at the synapse that open glutamate-gated ion-channels from the post-synaptic neuron shown in green. The opening of glutamate-gated ion-channels allows Na+ ions to enter the post-synaptic neuron, which depolarize the dendritic tree and flow towards the soma (2). As time passes, the accumulation of Na+ ions depolarizes the soma (3) until the membrane potential reaches about -45 mV. At this point, a large number of Na+ gated ion-channels present in the axon hillock (4) open, further depolarizing the cell and initiating an action potential that propagates down the axon (5) toward other neurons. The action potential is terminated through the opening of K+ ion-channels that bring back the neuron’s membrane potential to the resting value of about -64 mV. B: The membrane potential of the neuron is shown to change during an action potential (red). This change in membrane potential is due to the change in the number of open Na+ and K+ over time (magenta), where initially Na+ channels open, followed by K+ channels.

Now that the forms of the rate coefficients are given, it is useful to rewrite eq. (6) in a more meaningful way by dividing it with αn (V ) + βn (V ) :

In this way it is easier to notice that for a fixed voltage V, n approaches the limiting value n∞(V ) exponentially with a time constant τn (V ) which is easy to measure and fit to experimental data.

The second type of ion-channels involved in the generation of action potentials we’re going to discuss are the Na+ permeable channels. They are “more special” than the previously discussed K+ channels in that besides an activation gate they also have a blocking gate which makes these channels to be transiently opened upon depolarization with a probability:

Figure 12: Action potential in the Hodgkin-Huxley model. The first injected current pulse is too small for an action potential to occur while the second pulse is large enough to open Na+ channels and cause an action potential. For convenience the membrane potential has been shifted in the model by 65 mV such that Vm= 0 mV at rest instead of Vm= -65 mV.

While the Hodgkin-Huxley model captures the essence of neuronal action potential generation, in practice, neurons show a much larger diversity in the pattern of action potentials which is due to

the specific morphology of the dendritic tree and to other types of ion-channels that were omitted from the model (Fig. 13).

Figure 13: Neurons in the mammalian brain show widely varying electrophysiological properties that can be readily seen in the pattern of generated action potentials in response to a depolarizing current injection. A: Intracellular injection of a depolarizing current pulse in a cortical pyramidal cell results in a train of action potentials that slow down in frequency. This pattern of activity is known as “regular firing”. B: Some cortical cells generate bursts of three or more action potentials, even when depolarized only for a short period of time. C: Cerebellar Purkinje cells generate high-frequency trains of action potentials. D: Thalamic relay cells may generate action potentials either as bursts or E: as tonic trains of action potentials due to the presence of Ca2+ permeable ion-channels. F: Medial habenular cells generate action potentials at a steady and slow rate in a “pacemaker” fashion.

2.5 The Cable equation

In the previous section, the mechanism of action potential generation has been described using the Hodgkin-Huxley model, which assumes that the neuron has a somatic compartment containing Na+- and K+-permeable voltage gated ion-channels as well as passive leak channels. Obviously this simplified, yet very useful picture neglects the passive and active contribution of the dendritic tree that hosts a multitude of voltage-gated ion-channel. Examining again eq. (12) in the situation when a dendritic tree is added to the somatic compartment, several differences appear when injecting a current pulse in the soma. First, a part of the somatic injected current would “flow” in the dendritic tree which would slow down the charging of the somatic compartment. Second, the addition of the dendritic tree would change the passive properties of the cell by increasing its overall capacitance (due to more membrane) and by lowering the overall resistance (due to the additional leak channels in the dendritic membrane). Finally, the current injected in the soma and flowing in the dendritic tree can depolarize the dendrites and open other voltage-gated ion channels that could further depolarize the dendrites and lead to a locally generated action-potential. Other situations where the contribution of the dendritic tree has to be taken into account include the back-propagation of action potentials into the dendritic tree (Fig. 14A) after their generation in the axon-hillock (where the density of Na+-channels is the highest) and the propagation of excitatory synaptic potentials within the dendritic tree (Fig. 14B) that eventually reach the soma, depolarize it and produce action potentials.

Figure 14: Simultaneous intracellular recordings from soma and apical dendrite of cortical pyramidal neurons in slice preparations. A: A pulse of current was injected into the soma of the neuron to produce the action potential seen in the somatic recording. The action potential appears delayed and with smaller amplitude in the dendritic recording. B: A set of axon fibers was stimulated, producing and excitatory synaptic input. The excitatory postsynaptic potential (EPSP) is larger and peaks earlier in the dendrite than the soma. Note that the scale for potential is smaller than in A.

The attenuation and delay of signal propagation within neurons as shown in (Fig. 14A,B) is most severe when electrical signals travel down the long and narrow axons and dendrites. Due to the narrow cable like dimension of axon and dendrites, the radial variation of potential will be neglected compared to the longitudinal variations, and the membrane potential dynamics within such structures will depend on a single longitudinal spatial coordinate x and time: V(x,t).

When current flows through cellular compartments like axons and dendrites, due to the intracellular resistivity rL, a voltage gradient appears along a microscopic segment of length x and radius a. From this, the longitudinal resistance of such a cable segment can be calculated as:

In this situation, the voltage gradient V (x,t) =V (x + x,t) −V (x,t) along this segment would be

related to the current flow through Ohm’s law, with the sign convention that currents flowing in the direction of increasing x are positive:

Taking the limit x → 0 , the current flow at any point along a cable of radius a and intracellular resistivity rL in the direction of increasing x is:

To describe the membrane potential V(x,t) at a particular location along a thin segment, several other currents entering/leaving the segment have to be taken into account as shown in (Fig. 15).

Figure 15: A small cylindrical segment of a dendrite used in the derivation of the cable equation is shown together with several current sources/sinks that enter/leave the dendritic segment. (1) and (2) are the longitudinal currents at the boundary of the segment while (3) is the current charging up the membrane capacitance, (4) is a so called electrode current that represents an externally applied current to the segment e.g. via a patch pipette and (5) represents the sum of synaptic and membrane-bound ion-channel currents.

The form of the capacitive current (Fig. 15/3) can be understood from the definition of current flow

as a specific membrane capacitance / membrane area. The so called electrode current (Fig. 15/4) is just a mathematically convenient way of defining an externally injected current from a patch-clamp electrode that will be handy in the derivation of the cable equation and later will be replaced by simply Ie . The membrane current (Fig. 15/5) is the total current flowing through dendritic ion-

channels and depends on the detailed kinetics of each ion-channel present within the segment boundaries. Applying the principle of charge conservation (one of Kirchhoff’s laws) the following equation is obtained:

Let’s make a small mathematical trick by dividing eq. (28) by 2πa x so that it is easier to see that this is a second order differential equation with respect to the spatial coordinate x in the limit

x → 0 that contains the term:

Working under the assumption that the intracellular resistivity does not vary as a function of position, the cable equation is obtained:

Where the radius of the segment a is allowed to vary in the longitudinal direction, taking into account the situation of dendrites getting thinner as they are further away from the cell body. To solve this equation, boundary conditions have to be specified that describe what happens with the membrane potential at the ends of the segment considered, i.e. if the segment terminates or branches. In the case when the segment branches, the first boundary condition is set by the smooth variation of V(x,t) across the branching node such that V(x,t) of each branch has the same value at the branching point. The second boundary condition that must be applied in this case is the conservation of current flowing in/out of the branching node which according to eq. (27) sets the sum of terms containing ∂V (x,t) / ∂x to zero.

To gain more understanding into how membrane potentials propagate in axons and dendrites, we will make several (wild but instructive) assumptions that are hardly encountered in a real neuron: 1) the current injected in the segment produces only small membrane potential changes so that the membrane current im from eq. (30) can be approximated to change linearly with the membrane potential, i.e. im = (V −Vrest ) / rm ; 2) the radius a of the segment considered is constant. With these assumptions and making a variable change ν =V −Vrest that defines the membrane potential relative

to the resting potential, eq. (30) takes the simpler form:

If eq. (31) is multiplied now by rm then the factor that multiplies the time derivative turns into a membrane time constant τm = cmrm and the factor that multiplies the second order spatial derivative

will be arm / 2rL . With these substitutions, eq. (31) has the form:

with the membrane time constant:

and the electrotonic length:

spatial scale over which the segment membrane potential changes.

The steady state and transient solutions to eq. (32) are shown in (Fig. 16) in the infinite cable approximation. Although such an approximation is far from being realistic, it gives an idea of how dendrites attenuate and conduct brief synaptic current inputs. Realistic neuronal models, however, must take into account the specific dendritic morphology of the neuron, as well as the contribution of various dendritic ion-channels which can counteract the attenuation of signals in which case, the linear approximation of eq. (30) breaks down.

Figure 16: The solution to the cable equation is shown in the infinite cable approximation. A: Static solution for a constant electrode current where the potential decays exponentially away from the site of current injection. B: Timedependent solution for a delta function pulse of current. The potential is described by a Gaussian function centered at the site of current injection that broadens and shrinks in amplitude over time.

2.6 Multi-compartmental models

Neurons constantly receive a barrage of synaptic inputs, which can be thought of as brief current injections at various locations within the dendritic tree. Now that the cable equation was introduced in the previous section, it is easier to imagine how these “point” synaptic current injections depolarize various branches of the dendritic tree, eventually reaching the soma where they may produce action potentials.

The dendritic branches host several voltage-gated ion-channels that are important in shaping the membrane potential dynamics of dendrites in response to synaptic stimulation and realistic neuron models besides taking into account the morphology of the dendritic tree, must include this level of detail as well. Due to the complexity of the mathematical problem of describing how neurons process their synaptic inputs, computer models are used which are based on the numerical solution of eq. (30). This is done by discretizing eq. (30) which involves the division of a complex neuron into multiple compartments that are resistively coupled to each other (Fig. 17).