Neuronal Neuronal Circuits Circuits

CSCI 2323-1CSCI 2323-1

NeuronsNeurons

http://stepitup2007.org/article.php?list=class&class=20&offset=40

Neuron SchematicNeuron Schematic

http://faculty.etsu.edu/currie/excitation.htm

Action PotentialAction Potential

http://openwetware.org/wiki/BIO254:AP

CircuitsCircuits

http://www.physics247.com/physics-tutorial/parallel-circuits.shtml

http://en.wikipedia.org/wiki/Parallel_circuit

DefinitionsDefinitions Resistance: the opposition to the Resistance: the opposition to the

passage of a steady electric current passage of a steady electric current by a material.by a material.

Capacitance: the storage of energy Capacitance: the storage of energy due to separation of charge. due to separation of charge.

Current: flow of electric charge.Current: flow of electric charge. Potential: the energy released in the Potential: the energy released in the

transfer of a unit quantity of transfer of a unit quantity of electricity from one point to the electricity from one point to the other. other. All definitions provided by

Merriam-Webster Dictionary

Abbreviations to be usedAbbreviations to be used R: Resistance R: Resistance C: CapacitanceC: Capacitance I: CurrentI: Current V: PotentialV: Potential g: conductanceg: conductance

mm: membrane: membrane

ll: longitudinal (across the membrane): longitudinal (across the membrane)

ii: ion (in this case, Na: ion (in this case, Na++ , K , K++ & Cl & Cl-)-) a: radiusa: radius

Modeling the neuron as a Modeling the neuron as a circuitcircuit

AssumptionsAssumptions All ions contribute to V in a similar way.All ions contribute to V in a similar way. Only energy cost to the neuron is the Only energy cost to the neuron is the

separation of charge, without separation of charge, without distinguishing between the types of ions. distinguishing between the types of ions. Ex. NaEx. Na+/+/KK++ ATPase. ATPase.

Kirchoff’s First Law: Net charge can not Kirchoff’s First Law: Net charge can not pile up inside the individual circuit pile up inside the individual circuit elements. Thus, the volume of ions flowing elements. Thus, the volume of ions flowing = volume of ions flowing out.= volume of ions flowing out.

Cytosol is very resistant to the movement Cytosol is very resistant to the movement of electric charge.of electric charge.

[ ] and charge differences are only seen in [ ] and charge differences are only seen in a small region around the membrane.a small region around the membrane.

The membraneThe membrane

The Nernst RelationThe Nernst Relation

VViiNernstNernst=(-K=(-KBBT/ze)*ln([in]/[out])T/ze)*ln([in]/[out])

This equation tells us something This equation tells us something useful: the electric potential of our useful: the electric potential of our circuit with respect to an ion.circuit with respect to an ion.

We will see this equation A LOT!!!!We will see this equation A LOT!!!!

Resting Membrane Resting Membrane PotentialPotential

Donnan equilibrium helps us creates the Donnan equilibrium helps us creates the resting membrane potential (~72 mV). resting membrane potential (~72 mV). Created because ions have concentration Created because ions have concentration differences across the membrane.differences across the membrane.

ION PUMPS!!!!ION PUMPS!!!! Resting membrane potential is a steady Resting membrane potential is a steady

state, but not an equilibrium.state, but not an equilibrium. We can assume ∆V=VWe can assume ∆V=VNernst Nernst for all ions for all ions

present. Thus: we can get this equation:present. Thus: we can get this equation: [ ][ ]out, Na+out, Na+/[ ]/[ ]in, Na+in, Na+

= [ ]= [ ]out, K+out, K+ /[ ]/[ ]in, K+in, K+

==

[ ][ ]out, Cl-out, Cl-/[ ]/[ ]in, Cl-in, Cl-

All of this is equal to ∆V.All of this is equal to ∆V.

Ion fluxIon flux

∆∆V=VV=VNernst Nernst is boring.is boring.

jjq,i q,i is much more interesting, since it is much more interesting, since it tells us more about the flux of ions. tells us more about the flux of ions.

Ions crossing the membranes lead to Ions crossing the membranes lead to cool stuff happening (Action cool stuff happening (Action Potentials!).Potentials!).

Ohm’s LawOhm’s Law

Ohm’s Law: ∆V=IOhm’s Law: ∆V=IiiRRii+V+ViiNernstNernst

∆∆V=Vin-VoutV=Vin-Vout Potential difference across the membranePotential difference across the membrane

IIii=j=jq,iq,iAA Current generated by ions crossing the Current generated by ions crossing the

membranemembrane

RRii=1/g=1/giiAA Resistance of the membrane to each ion.Resistance of the membrane to each ion.

CapacitorsCapacitors

As before, the membrane is multi-As before, the membrane is multi-purpose.purpose.

Net charge across the membrane is Net charge across the membrane is zero. Thus, we can’t have charge zero. Thus, we can’t have charge imbalance (DONNAN EQUILIBRIUM).imbalance (DONNAN EQUILIBRIUM).

However, charge can accumulate in a However, charge can accumulate in a very small region around the very small region around the membrane. This allows for charge to membrane. This allows for charge to flow toward the membrane.flow toward the membrane.

q=C(∆V) q=C(∆V) d(∆V)/dt=I/C d(∆V)/dt=I/C

GOOD!!!GOOD!!!

We have been able to successfully We have been able to successfully create several of the parameters create several of the parameters modeling the neuronal circuit, modeling the neuronal circuit, including the potential difference including the potential difference across the membrane, the across the membrane, the capacitance. All of this has been capacitance. All of this has been thanks to the innate physiology!thanks to the innate physiology!

BUT NOT GOOD BUT NOT GOOD ENOUGH!!!ENOUGH!!!

Our current model fails to accurately Our current model fails to accurately model a real axon. Our model assumes model a real axon. Our model assumes that the axon is small (not always the that the axon is small (not always the case), or a large membrane maintained at case), or a large membrane maintained at a potential that is uniform across its a potential that is uniform across its length. Real axons don’t work like this, length. Real axons don’t work like this, since only small parts of the axon are since only small parts of the axon are actually propagating the action potential.actually propagating the action potential.

Some equations just don’t cut it. Some equations just don’t cut it. So, we need a little more advanced So, we need a little more advanced

formula to model what is really going on.formula to model what is really going on.