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**Instructor: Dr Sachin S. Talathi**

BME 6938 Neurodynamics Instructor: Dr Sachin S. Talathi

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**Recap Neurons are excitable cells Neuronal classification**

Communication in the brain is mediated by synapses: Electrical and Chemical

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Neuronal Signaling Neuronal signaling is mediated by the flow of dissociated ions across the cell membrane. The fundamental laws that govern the flow of these ions through the cell membrane are: Ficks Law of Diffusion Ohms Law of Drift Space charge neutrality Einsteins Relation between diffusion and drift

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**Fick’s Law of Diffusion**

Fick’s law relates the diffusion gradient of ions to their concentration. Jdiff: Diffusion flux, measuring the amount of substance flowing across unit area per unit time (molecules/cm2s ) D: Diffusion coefficient ( cm2/s ) [C]: Concentration of the ions (molecules/cm3 ) The negative sign indicates the flow of ions across decreasing concentration gradient Ficks Law Animation

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**Ohms law of drift (Microscopic view)**

Charged particle in the presence of external electrical field E experience a force resulting in their drift along the E field gradient Jdrift: Drift flux, measuring the amount of substance flowing across unit area per unit time ( molecules/cm2s ) mu: electrical mobility of charged particle(cm2/sV) [C]: Concentration of the substance (ions) (molecules/cm3 ) z: Valence of ion The negative sign indicates the flow of ions across decreasing voltage gradient. The ion flux depends on electrical resistance which in turn depends on ion mobility

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**Space charge neutrality**

Biological systems are overall electrically neutral; i.e., the total charge of cations in a given volume of biological material equals the total charge of anions in the same volume biological material

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Einstein relation It relates the diffusion constant (effect of motion due to concentration gradients) of an ion to its mobility (effect of motion due to electrical forces) Basic idea is that the frictional resistance created by the medium is same for ions in motion due to drift and diffusion.

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**Some high-school chemistry**

1 mole= Avogadro’s number (NA) of basic units (atoms, molecules, ions…) Concentration is typically given in units of molar. 1 Molar=1 mole/litre=10-3 moles/cm3 Relation between gas constant (R) and Boltzmann’s constant (k): R=kNA Faraday constant F: Magnitude of one mole of charged particles: F=qNA

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Some-algebra Membrane capacitance of a cell membrane is around 1 microF. Concentration of ions within and outside of a cell is 0.5 M. Determine the fraction of free (uncompensated) ions required on each side of a spherical cell of radius 25 micro m to produce 100mV? Ans: ~ % For realistic cell dimension, from above calculations we see that generation of 10s of mV of voltage does not violate space-charge neutrality (~99.9% of charges are compensated)

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**Nernst-Plank Equation**

Nernst Plank equation governs the current generated from the flow of individual ions across the cell membrane. Continuity Equation The time dependent Nernst Plank Equation: Derive the eq in class

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Nernst Equation Nernst equation is special case of NPE, where in the membrane potential is obtained as a function of concentration gradient across the cell membrane when the net current flow generated by the ion is zero.

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**Typical scale of reversal potential values**

Question: What is the direction of flow of following ions under normal conditions? 1.Na+ 2. K+ 3. Ca2+ 4. Cl- (Hint: Look at the chart of reversal potentials and Nernst Equilibrium potential equation) At 37 oC The negative sign indicates the flow of ions across decreasing concentration gradient mV

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**Specific Example Ion concentration for cat motoneuron: Vm=-70 mV**

Nernst Potential: At body temperature 37oC Inside mol/m3 Outside mol/m3 Na+ 15 150 K+ 5.5 Cl- 9 125 Specific problem: The resting potential is not equal to reversal potential for Na and K; what does that mean?? How is resting potential achieved?

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**Ion distribution and Gradient maintenance**

Active Transport: Flow of ions against concentration gradient. Requires some form of energy source Examples: Na+ pump Read section in Johnston’s& Wu book for more information. Passive Transport: Selective permeability to some ions results in concentration gradient No energy source required Passive distribution of ions can be determined using the Donnan rule of equilibrium Ion pumps are proteins inside the plasma membrane of cells, which pump ions across the plasma membrane often against the concentration gradient. Read sectio

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**Donnan Equilibrium Rule**

The membrane potential equals the reversal potential of all ions that can passively permeate through the cell membrane. Mathematically the Donnan Rule implies: Have a look at Donnan Rule in works; through animaltion developed by Larry Keeley:

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**Graphical illustration of ion gradient maintenance**

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**Example: Application of Donnan Rule**

Consider a two compartment system separated by a membrane that is permeable to K+ and Cl- but is not permeable to a large anion A-. The initial concentrations on either side of membrane are: Is the system in electrochemical equilibrium (no ion flow across the membrane? If not, what direction the ions flow? And what are the final equilibrium concentrations? Ion type I (conc in mM) II (conc in mM) A- 100 K+ 150 Cl- 50

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**Steady state solution to NPE**

We can solve the NPE equation in steady state (ie no time dependence for concentrations). A special case was seen through NE, wherein in addition to the membrane being in steady state, the membrane was in equilibrium (I=0).

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**Boundary conditions-Cell membrane**

Permeability P: Defined as absolute magnitude of ion flux when there is a unit concentration difference between internal and external fluids and the concentration is linear function of distance within the membrane (in molar units) : Partition coefficient, which measures the drop in concentration of species at fluid membrane interface.

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**Constant field assumption**

Most common assumption is charge density within the membrane is identically zero. This assumption leads to the following expression for V(x) with boundary conditions: V(x=0)=Vm and V(x=l)=0 We get the following expression for current: where Note: Poisson equation from Electrostatics:

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**Goldman Hodgkin Katz Model**

It is widely used model to predict the resting membrane potential Vm for nerve cells The formula for Vm is derived as a solution NPE with constant field assumption Further more it is assumed that ions flow across the cell membrane without interacting with each other For membrane that is permeable to M positive monovalent ions and N negative monovalent ions, the GHK formula for Vm in equilibrium conditions is: Derive this equation in class

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**Exercise (extra credit)**

Show that the resting equilibrium membrane potential Vm for a cell membrane that is permeable to monovalent and divalent ions is given by

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**Application of the GHK equation**

Lets use GHK eqn to determine the contribution to membrane potential from active ion transport mechanism’s. Na-pump result in flow of 3 Na+ ions across the cell membrane for every 2K+ ions. What is the resulting equilibrium potential of the cell of squid axon for which the concentration gradients across the cell are: The permeability ratio is Pk:Pna=1:.03 Ion type Inside (mM) Outside (mM) K+ 400 20 Na+ 50 550

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BME 6938 Mathematical Principles in Neuroscience Instructor: Dr Sachin S. Talahthi.

BME 6938 Mathematical Principles in Neuroscience Instructor: Dr Sachin S. Talahthi.

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