co6 - neurons - lecture 1

8/12/2019 CO6 - Neurons - Lecture 1

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NEURAL ELECTRICITY

Neurons are cells that process and transmit information, mainly under the form of stereotypedelectrical impulses, action potentials or spikes. A typical neuron (above, a pyramidal cortical cell)has a soma, an axon and a dendritic tree. The soma contains the genetic material of the cell. Thedendrites receive inputs from other neurons. Spikes are produced at the axon initial segment,near the soma, and electrically transmitted along the axon. The axon makes synapses with targetneurons, named postsynaptic neurons. A synapse is a small space between two neurons. When aspike arrives at a synapse, neurotransmitters are released and produce electrical currents in thepostsynaptic neuron. Excitatory neurons generally make synapses on dendritic spines. Neuronscan also make synapses on the soma or even sometimes on the axon. The synaptic currents thenpropagate to the soma, which changes the electrical potential at that point. When the potential islarge enough, a spike is produced at the axon initial segment and is then propagated along theaxon.



Polarization and membrane potential

The neuron is delimited by a very thin bilipidic membrane (about 2 nm thick). Outside themembrane is the extracellular milieu, similar to sea water: it contains many sodium (Na+) and

chloride (Cl‐) ions. Ions carry electrical charges, responsible for neural electricity. On the insideis the intracellular milieu, which has a different ionic composition, with many potassium ions(K+). In the membrane, there are proteins called ionic channels, which make tiny holes that letspecific types of ions pass. At rest (without any stimulation), channels are mostly permeable toK+ ions ‐ these are often called leak channels. Ions move randomly by diffusion, under the effectof temperature. It produces a flux of K+ ions that leave the cell, proportional to the gradient ofconcentration (Fick’s first law of diffusion). Diffusion of K+ ions creates an excess of positivecharges outside the cell and an excess of negative charges inside the cell, which produce anelectric field across the membrane. An electric field represents the force exerted on electricalcharges, because of the presence of other charges. Thus this electric field across the membrane

pushes K+ ions back to the intracellular space. Equilibrium is reached when the flux of ions dueto the electric field exactly matches the flux due to diffusion.At equilibrium, there is an electric field directed inwards. The electric field E is the opposite ofthe gradient of electrical potential: . Thus, the potential V decreases along the electricfield: the intracellular potential Vi is lower than the extracellular potential Ve. The membrane

potential is defined as Vm = Vi – Ve and Vm<0, typically Vm ≈ ‐70 mV. The membrane is polarized .An increase in Vm is a depolarization (|Vm| decreases), a decrease in Vm is a hyperpolarization.The potential at equilibrium is called the resting potential and is often denoted EL (L for leak) orV0.



The Nernst potential

At equilibrium, the flux of ions due to the electric field exactly compensates diffusion. Themembrane potential is then equal to the Nernst potential or equilibrium potential:

log

where R is the gas constant, F is the Faraday constant, T is the absolute temperature (in Kelvin),z is the ion charge (z = 1 for K+, z = ‐1 for Cl‐), and [K+]ext and [K+]int are the extra‐ andintracellular concentrations of K+ ions. A very small number of ions need to be displaced toestablish the Nernst potential and therefore, assuming that extra‐ and intracellular volumes arelarge, concentrations are generally assumed to be unchanged by the process (but this is anapproximation).

Typical equilibrium potentials in mammalian cells at 37°C K+ ‐90 mVNa+ 60‐90 mVCl‐ ‐90 mVCa2+ 136‐145 mV



*A dynamic equilibrium

The Nernst potential of K+ is typically about ‐90 mV, but the resting potential in a cell isgenerally about ‐70 mV. This is because ionic channels are also permeable to other ionic species.In this case, the equilibrium potential depends on the relative permeability of the membrane to

the different ionic species. It is given by the Goldman‐Katz formula:

log

where Pi is the membrane permeability to ionic species i. The formula generalizes to differentionic compositions. The Nernst potential is specific of an ion, but the G‐K potential is specific of aionic channel.At equilibrium, the net flux of charges is zero overall, but not for each ionic species: K+ ions flowout of the cell and Na+ ions flow into the cell. This is a dynamic equilibrium. For this reason,channels responsible for the resting potential are often called leak

channels.

If no mechanism keeps the ionic concentrations constant, these tend to equalize and themembrane potential vanishes (Vm = 0). Such a mechanism is provided by the Na+/K+ pump. Thisis an enzyme in the membrane which exchanges Na+ ions for K+ ions against their concentrationgradients, which requires energy in the form of ATP, the energy currency of cells. Most of energyconsumption of the brain is spent by this pump.The Na+/K+ pump exchanges 3 Na+ ions against 2 K+ ions. Because of this asymmetry, itproduces a small negative (or outward ) current, which is why it is sometimes called anelectrogenic pump. This is usually ignored in models because this current is small and slow, but itnevertheless constitutes an adaptive current, particularly in response to action potentials, whichproduces large inward fluxes of Na+ ions.



The equivalent electrical circuit

When a synapse is activated, it produces a current. A current I can also be injected into the somathrough an electrode (I>0 when positive charges enter the cell). The membrane is a very thinelectrical insulator, and can be modeled as a capacitor. The capacitive current through a patch ofmembrane equals IC = C.dVm/dt, where C is the capacitance, proportional to the area A of thepatch (Ic>0 when positive charges leave the cell). That is, C = A x cm, where cm is the specific

membrane capacitance. This value, defined per unit area of membrane, is essentially constantacross neurons: cm ≈ 0.9 µF/cm².A current IL also flows through the leak channels. This current is zero when V m equals the restingpotential EL (L for leak), it is positive when Vm>EL and negative when Vm<EL. That is, IL has thesame sign as (V‐EL). A linear approximation is IL = gL(V‐EL), where gL is a constant. It is generallyconsidered a good approximation. This is electrically equivalent to the current flowing through aresistor with resistance R = 1/gL, in series with a battery EL. We call R the membrane resistanceand gL the leak conductance. Thus a patch of membrane is electrically equivalent to the electricalcircuit above.



The membrane equation

We make the assumption that the cell is isopotential , that is, that the membrane potential isidentical everywhere in the cell, and we consider currents through the entire membrane. Acurrent I is injected into the cell, which reflects the current coming from an intracellularelectrode or a synaptic current. By Kirchhoff's law, the injected current equals thetransmembrane current: I = Ic +IL. We obtain the membrane equation:

A common equivalent form is:

where R=1/gL is the membrane resistance and τ = RC is the membrane time constant . When aconstant current is injected, the stationary value of Vm is . When the current isswitched from 0 to I (step current), Vm approaches the stationary value with characteristic timeτ. That is, the distance between Vm and is divided by e (≈2.7) within a time τ. The solution tothe differential equation is:

/



The integrate‐and‐fire model

The integrate‐and‐fire model consists of a membrane equation (“integrate”) and an explicit

firing condition (“spike”): when Vm

reaches a threshold Vt

, a spike is produced. The potential isthen instantaneously reset to a value Vr. In its simplest form, the equations are:

when ∶ → Typical values are Vt = ‐55 mV and Vr = ‐70 mV. This model is a phenomenological description ofthe action potential, in that spikes are explicitly introduced rather than being the result of thedynamics of biophysical equations, as in the Hodgkin‐Huxley model.

The threshold current or rheobase is the smallest constant current I that makes the neuron firerepetitively. If the neuron does not fire, the stationary voltage is , which has to be smallerthan Vt. Therefore the rheobase is:

∗



The integrate‐and‐fire model: firing rate

The firing rate is the average number of spikes per unit time. It equals the inverse of the averageinter‐spike interval (ISI). For a sequence of spikes at times t1, ..., tn, the ISI is Ti=ti+1‐ti. For aconstant current I, the ISI is the time T for a solution of the membrane equation starting from V r to reach threshold Vt:

log

The firing rate is F=1/T. The relationship between constant current and firing rate is the current ‐

frequency relationship.



Synaptic currents

The axonal terminal at a synapse contains vesicles with neurotransmitters. When a spike arrivesat a synapse from a presynaptic neuron, vesicles fuse with the membrane and release theirneurotransmitters. These molecules bind with receptors on the postsynaptic membrane, whichopens specific ionic channels. Ions enter and produce a current Is(t) on the postsynaptic side.The membrane equation becomes:



A simplified synapse

We consider that the channels open and close very fast when a presynaptic spike arrives, and leta total charge Q enter the postsynaptic neuron. The membrane potential then changes by Q/C.

The resulting time course of Vm is called the postsynaptic potential

(PSP), and is described by thefollowing equations:

→ at spike time

Equivalently, the synaptic current can be modeled as , where is the Diracfunction, a pseudo‐function that equals zero at every time t≠0, and with a unit integral (=1).Thus the total charge is . The equivalent equation for the PSP is then:



A more realistic synapse

More realistically, the principles of electrodiffusion also apply to synaptic currents:

where gs(t) is the conductance of the ionic channels and Es is their equilibrium potential. Thesynaptic conductance increases when the neurotransmitters bind to the receptors and ionicchannels open. It decreases when the channels close. The membrane equation becomes:



Excitation and inhibition

The effect of a presynaptic spike on Vm depends on the driving force (Es‐Vm): it is depolarizing ifEs>Vm and hyperpolarizing if Es<Vm. This depends both on the synapse properties (equilibriumpotential Es) and on the postsynaptic membrane potential Vm.A synapse is considered excitatory if its activation makes the postsynaptic neuron moreexcitable, that is, if it lowers the rheobase. It is considered inhibitory if it raises the rheobase. Therheobase current I* is such that at threshold Vt, all non‐capacitive currents sum to zero (that is,dVm/dt=0). The synaptic current at Vt is . When Es>Vt, this is positive. This means thata lower current I* is required to reach threshold: the synapse is excitatory. When E s<Vt, thesynapse in inhibitory.The threshold Vt is typically about ‐55 mV. In the table below, glutamate receptors are thusexcitatory, GABA receptors are inhibitory. According to Dale's principle, a neuron expresses only

one type of neurotransmitter in its axonal terminals. Therefore, there areexcitatory neurons

andinhibitory neurons.

Inhibition can be silent or shunting if the reversal potential Es equals the resting potential EL. Inthis case, activating the synapse at rest has no visible effect on Vm. Inhibition can even bedepolarizing, when EL<Es<Vt: activating the synapse at rest increases Vm, although it makes cellless excitable. Excitation is always depolarizing, because Vt>EL.

Major synapse types

Neurotransmitter

Receptor

Es PropertiesGlutamate AMPA 0 mV fastNMDA 0 mV slow, voltage‐dependent

GABA GABA‐A ‐70 mV fastGABA‐B ‐100 mV slow

Note that the values of Es are only indicative. There are many other neurotransmitters, forexample glycine and acetylcholine.



Synapse kinetics

The membrane contains many ionic channels. At any given time, a ionic channel is in one of anumber discrete states. A simple model consists of two states, open (O) and closed (C). Ions canpass through the channel only in the open state. The channel switches randomly from one stateto another at some rate, the expected number of transitions per second. The opening ratedepends on the neurotransmitter concentration [L] (L for ligand): it equals α.[L]. The closingrate β is constant. With many channels, the proportion of open channels P(t) is governed by the

following equation: 1

An approximation when the neurotransmitter is present during a very short time is:

→ at spike time

The approximation is valid when a small proportion of channels are open. If gmax is the maximumsynaptic conductance, when all channels are open, the membrane equation becomes:

→ at spike timewhere τs=1/β is the synaptic time constant . This type of model is named conductance‐based

model .More complex synaptic models can be obtained, for example by considering more states. Otheraspects of synaptic dynamics include probabilistic transmission and plasticity.



Synaptic integration: temporal integration

When several spikes are received at a synapse, their effects on the synaptic conductancecombine. In the simplest model, this combination is linear:

→ at spike timeIndeed if gs*(t) is the time course of the synaptic conductance for one spike at time t=0, then theconductance for a sequence of spikes at times t1,..., tn is:

∗

This is a linear superposition principle. To prove this principle, observe that gs and gs* follow thesame differential equation, because it is linear. At spike time ti, gs indeed increases by .Therefore the membrane equation can also be written as follows:

∗

The same superposition principle applies to the idealized synapse model with constant chargetransfer.



Synaptic integration: spatial integration

The synaptic currents produced by spikes received at different synapses sum. Synapses mayhave different properties, for example equilibrium potential Ei. With n synapses, the equationsare:

, 1

→ on spike at synapse iA special case is when the synapses have the same properties and the kinetics are linear (lineardifferential equations and additive update). We can then define a lumped variable gs as the total

conductance over all synapses:

The linear superposition principle applies to the lumped variable. The equations become:

→ on spike at synapse iIf there are k types of synapses, we may define k lumped variables in the same way.We may also apply the superposition principle to describe the dynamics through a singleequation:

∗

where tij is the time of spike j at synapse i, and ∗ is the time course of the synapticconductance for one spike at time t=0 at synapse i.



Current ‐based models

The equations of conductance‐based models are non‐linear, because the membrane equationcontains products of variables (gs(Es‐Vm)). They can be linearized by assuming that the drivingforces (Es‐Vm) are constant. This amounts to replacing Vm by a constant value, equal to theaverage membrane potential V0, or to the resting potential. The quality of this approximationdepends on the variability of (Es‐Vm). Given that Vm varies essentially between EL and Vt, theapproximation is valid when Es>>Vt or when Es<<EL. Therefore, the approximation is typicallymore reasonable for excitatory synapses than for inhibitory synapses.This amounts to replacing synaptic conductances by synaptic currents, hence the name current ‐

based models. The equations become:

, 1

→ on spike at synapse iwhere and . As before, the linear superposition principle applies,and we can describe synapses with identical properties using lumped variables.



The post ‐synaptic potential

The post‐synaptic potential (PSP) is the time course of V m in response to a single spike at a givensynapse. For current‐based models with linear synaptic equations and additive synaptic updates

( → ), the entire differential system is linear and therefore the linear superpositionprinciple applies. This means that the response of the neuron to a sequence of spikes can bedescribed as:

,

where PSPi(t) is the time course of the PSP at synapse i, and tij is the timing of spike j at synapsei.This is the integral form of the neuron model, as opposed to the differential form. It must be

stressed that expressing this integral form as a sum relies on the assumption of the linearsuperposition principle, which only holds under restrictive conditions.



Electrical synapses

Neurons can be connected directly through gap junctions or electrical synapses. The membranesof the two cells are then in direct contact, and a current can flow between them (as well as somemolecules). Most gap junctions can be electrically modeled as a resistor. The current flowing

from neuron B to neuron A is then IBA=(VB‐VA)/Rg, where Rg is the resistance of the gap junction.The membrane equations of neurons A and B are then:

where the two neurons were assumed to have the same properties.

co6 - neurons - lecture 1

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