1. PETER PAZMANY CATHOLIC UNIVERSITY
PETER PAZMANY
CATHOLIC UNIVERSITY
SEMMELWEIS
UNIVERSITY
Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**
Consortium leader
PETER PAZMANY CATHOLIC UNIVERSITY
Consortium members
SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER
The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***
**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben
***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.  
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2. INTRODUCTION TO FUNCTIONAL NEUROBIOLOGY
Peter Pazmany Catholic University
Faculty of Information Technology
BEVEZETÉS A FUNKCIONÁLIS NEUROBIOLÓGIÁBA
INTRODUCTION TO FUNCTIONAL NEUROBIOLOGY
By Imre Kalló
Contributed by: Tamás Freund, Zsolt Liposits, Zoltán Nusser, László Acsády, Szabolcs Káli, József Haller, Zsófia Maglóczky, Nórbert Hájos, Emilia Madarász, György Karmos, Miklós Palkovits, Anita Kamondi, Lóránd Erőss, Róbert Gábriel, Kisvárdai Zoltán
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3. Introduction to functional neurobiology: Neuronal modelling
Neuronal modelling
Szabolcs Káli
Pázmány Péter Catholic University, Faculty of Information Technology
Infobionic and Neurobiological Plasticity Research Group, Hungarian Academy of Sciences – Pázmány Péter Catholic University – Semmelweis University
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4. Descriptive – What is it like? Mechanistic – How does it function?
Neural modeling
Types of models:
Descriptive – What is it like?
Mechanistic – How does it function?
Explanatory – Why is it like that?
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5. Some fundamental questions
How is information encoded by action potential trains?
Neurons respond to input typically by producing complex spike sequences that reflect both the intrinsic dynamics of the neuron and the temporal characteristics of the stimulus.
Simple way: Count the action potentials fired during stimulus, repeat the stimulus and average the results:
Picture: Recordings from the visual cortex of a monkey. A bar of light was moved through the receptive field of the cell at different angles (figure A). The highest firing rate was observed for input oriented at 0 degrees (figure B).
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6. Some fundamental questions
How to decode information encoded by action potential trains?
Example: Arm movement position decoding: If we take the average of the preferred directions of the neurons weighted by their firing rates, we get the arm movement direction vector.
Picture: Comparison of arm position and arm position-sensitive neurons. The population activity was recorded in 8 directions. Arrows indicate vector sums of preferred directions, which is approximately the arm movement direction.
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7. Some fundamental questions
Why does a given part of the brain use a specific type of coding? Example: Visual input noise filtering in ganglion cells: The structure of the receptive field changes according to the input signal-to noise ratio.
Solid curves are for low noise input (bright image), dashed lines are high noise input. Left: The amplitude of the predicted Fourier-transformed linear filters. Right: The linear kernel as a function of the distance from the center of the receptive field
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8. Some fundamental questions
How do neurons as information processing units function?
Specifically, what is the relation between the temporal and spatial pattern of the input and the spatial and temporal pattern of the output?
Example:
Picture: the effects of constant sustained dendritic current injection in a hippocampal pyramidal cell. The cell responds with a burst of spikes, then sustained spiking. In distal regions only a slow, large-amplitude initial response is visible, corresponding to a dendritic calcium spike.
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9. Some fundamental questions
How do neurons communicate, and what collective behaviors emerge in networks? Example: Orientation selectivity and contrast invariance in the primary visual cortex
Left picture: schematics of a recurrent network with feedforward inputs.
Middle picture: The effect of contrast on orientation tuning. Figure A: orientation-tuned feedforward input curves for 80%,40%,20%,10% contrast ratios.
Right picture: The output firing rates for response to input in figure A.
Due to network amplification, the response of the network is much more
strongly tuned to orientation as a result of selective amplification by the recurrent network, and tuning width is insensitive to contrast.
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10. Some fundamental questions
How does cellular-level (synaptic) plasticity function? How can we understand behavioral-level learning? What is the connection between the two? Example: The development of ocular stripes in the primary visual cortex
Left picture: schematics of the model network where right- and left- eye inputs from a single retinal location drive an array of cortical neurons.
Right picture: Ocular dominance maps, the light and dark areas along the cortical regions at the top and bottom indicate alternating right- and left-eye innervation. Top: In vitro measurements. Bottom: The pattern of innervation for the model after Hebbian development.
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11. Lipid bilayer (insulator) Ion channels Selectivity Modulation:
Basics
Neuronal membrane:
Lipid bilayer (insulator)
Ion channels
Selectivity
Modulation:
Membrane potential
Intracellular messengers (for example Ca2+)
Neurotransmitters and -modulators
Ion pumps
Receptors not bound to ion channels
Others
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12. Elementary physical laws 1: Ohm's law for drift
: Drift flux
: Mobility z
: Valence of the ion
[C] : Concentration of ions
Charged particles (e.g., ions) in a fluid (e.g., cell plasma or extracellular fluid) experience a force resulting from the interaction of their electric charges and the electric field in the environment.
This equation states that drift of positively charged particles takes place down the electric potential gradient and is everywhere directly proportional to the magnitude of that gradient.
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13. D : Diffusion coefficient [C] : Concentration of ions
Elementary physical laws 2: Fick's law, diffusion of particles caused by concentration differences
: Diffusion flux
D : Diffusion coefficient
[C] : Concentration of ions
Fick's law states that diffusion takes place down the concentration gradient and is everywhere directly proportional to the magnitude of that gradient, with proportionality constant D.
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14. D : Diffusion coefficient k : Boltzmann constant
Elementary physical laws 3: The Einstein relation between diffusion and mobility
 : Mobility
D : Diffusion coefficient
k : Boltzmann constant
T : Temperature (Kelvin)
q : Charge of the molecule
This relationship states that diffusion and drift processes in the same medium are additive, because the resistances presented by the medium to the two processes are the same. This equation enables us to convert the diffusion coefficient to mobility.
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15. Nernst-Planck equation
From Ohm's law, Fick's law, and the Einstein relation, the total current density is:
which is the Nernst-Planck equation (in molar form, J is in mol/sec·cm2, and u=µ/NA, where NA is Avogadro’s number (6x1023/mol)).
The current density form of the equation can be obtained by multiplying the molar flux (J) by the molar charge (zF):
where I is A/cm2. The Nernst-Planck equation describes the ionic current flow driven by electrochemical potentials (concentration gradient and electric field). This equation describes the passive behavior of ions in biological systems.
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16. If the total electric current across the membrane is zero (I=0),
Nernst equation
If the total electric current across the membrane is zero (I=0),
If we integrate both sides across the membrane, and define the membrane potential of a cell as
the reversal potential of ion i, defined as the membrane potential where the membrane current carried by ion i is zero, can be expressed as:
This is called the Nernst equation.
The Nernst-equation also implies that when the membrane is at the reversal potential of an ion species, the cross-membrane voltage (drift force) and concentration gradient (diffusion force) exert equal and opposite forces.
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17. Goldman-Hodgkin-Katz equation
In a typical neuron
but
and
The anions inside the cell, which cannot pass through the membrane, must be taken into account in the calculation, too.
Most of the ions are not in equilibrium:
ENa: 50 mV EK: -90 mV
ECa: 150 mV ECl: -70 mV
Thus, ionic currents start to flow as soon as ion channels open, and ion concentrations must be maintained by active transport (ion pumps).
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18. Goldman-Hodgkin-Katz equation
The equilibrium membrane potential is determined by the ion permeabilities of the membrane (Goldman-Hodgkin-Katz equation):
Where
Pion is the permeability for that ion (in meters per second)
V is the membrane potential
[ion] is the concentration of that ion (in moles per cubic meter)
The Goldman-Hodgkin-Katz equation is used in cell membrane physiology to determine the equilibrium potential of the cell's membrane taking into account all of the ions that are permeant through that membrane.
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19. The following processes influence the membrane potential:
Resting potential
The resting potential (Vrest) is between -80 and -50mV, and with the change of the permeabilities the membrane potential can take up values between -90mV and +50mV.
The following processes influence the membrane potential:
Depolarizing and hyperpolarizing voltage-gated conductances
Excitatory and inhibitory synapses
Shunting effect: Conductances with reversal potentials near the resting potential (for example Cl-), may pass little net current. Instead, their primary impact is to change the membrane resistance of the cell. Such conductances are called shunting, because they increase the total conductance of a neuron.
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20. Passive, isopotential (single compartment) neuron model
V : Membrane potential [V].
Ie : Current injected into the cell, with an electrode for example [A].
Q : Excess internal charge [C].
Rm : Membrane resistance, treated as a constant in the equations (specific membrane resistance, rm) [Ohm].
Cm : Membrane capacitance, treated as a constant in the equations (specific membrane capacitance, cm) [F].
The cell membrane is represented by a resistance and a battery in parallel with a capacitance.
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21. 𝐶 𝑚 𝑑𝑉 𝑑𝑡 =− 𝑔 𝑚 𝑉− 𝐸 𝑟 + 𝐼 𝑒 Calculating the membrane current 1
From the definition of capacitance, the amount of current (charge per unit time) needed to change the membrane potential of a neuron with a total capacitance Cm at a rate dV/dt is:
Because of the principle of conservation of charge, the time derivative of the charge dQ/dt is equal to the current passing into the cell, so
Where Er is the resting potential of the cell. In most equations membrane conductance (gm) is used instead of resistance (gm=1/Rm ), because it is directly related to biophysical properties of the neuron:
𝐶 𝑚 𝑑𝑉 𝑑𝑡 =− 𝑔 𝑚 𝑉− 𝐸 𝑟 + 𝐼 𝑒
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22. Calculating the membrane current 2 : The membrane time constant
The product of the membrane capacitance and the membrane resistance is
a quantity in units of time, called the membrane time constant, denoted by tau:
The membrane time constant sets the basic time scale for changes in the membrane potential and typically falls in the range between 10 and 100 milliseconds.
The total membrane conductance can change dynamically, causing the membrane time constant to change, too.
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23. Response to current step at t=0
Example:
(ΔV = 100 mV, tau = 20msec, V0 = -70 mV)
The passive membrane behaves as a low-pass filter.
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24. Total current flowing through the membrane
If there are multiple conductances (ion channels):
Where gi(V-Ei) is the current flowing through ion channel i.
Equivalent electric circuit:
Ion channels and synaptic channels can be represented as variable conductances.
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25. Describing extended neurons I : the cable equation
The membrane potential can vary considerably over the surface of the cell membrane:
Figure A: The delay and attenuation of an action potential as it propagates from the soma out to the dendrites of a cortical pyramidal neuron.
Figure B: The delay and attenuation of an excitatory postsynaptic potential (EPSP) initiated in the dendrite by synaptic input as it spreads to the soma.
Dendritic and axonal cables are usually narrow enough that variations of the potential in the radial direction (at a given axial location) are negligible compared to longitudinal variations, thus we only need a single longitudinal coordinate, denoted by x. 25

26. Describing extended neurons I : the cable equation
Cable theory uses mathematical models to calculate the flow of electric current (and accompanying voltage) along passive neuronal fibers (neurites), particularly dendrites. 26

27. Describing extended neurons I : the cable equation
Simplification: Represent the dendrite as a finite length cable.
Subdivide the dendrite into little pieces, small enough that the voltage is approximately constant everywhere within each such piece.
Membrane capacitance is denoted by CM
Membrane resistance is denoted by RM
Longitudinal (axial) resistance is denoted by RA 27

28. V1, V2: Membrane potential at the ends of the cable (mV).
Axial resistance (RA)
L: Cable length (cm).
d: Cable diameter (cm).
V1, V2: Membrane potential at the ends of the cable (mV).
Ra: Axial (or longitudinal) resistance (ohm)
RA: Specific axial resistance
IA: Longitudinal current (A)
Axial resistance is proportional to the length of the segment (long segments have higher axial resistances than short ones). It is inversely proportional to the cross-sectional area of the segment (thin segments have higher resistances than thick ones). 28

29. Longitudinal current (IA)
For the cylindrical segment of dendrite shown in the figure, the longitudinal current flowing from right to left satisfies V2-V1 = IARA (Ohm’s law). This can be rewritten as:
If we take the limit of this expression for infinitesimally short cable segments, the equation becomes a partial differential equation: 29

30. The segment of neuron used in the derivation of the cable equation
(cm is the membrane capacitance):
We divide the membrane into small cylindrical parts. One cylinder of the membrane has a surface area of and hence a capacitance of 30

31. Which is called the cable equation.
The total longitudinal current entering the cylinder is the difference between the current flowing in on the left and that flowing out on the right, so the current balance equation becomes: If
Which is called the cable equation.
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32. Synaptic currents are ignored
Linear cable equation
To solve the cable equation we need to know the membrane current. The simplest (linear) case, which can be solved analytically, is the following:
Synaptic currents are ignored
Membrane current is a linear function of the membrane potential
In other words we are looking for the value of Vm(x,t) given Im(x,t)
In real neurons, a linear approximation for the membrane current is valid only if the membrane potential stays within a limited range. Then the membrane current per unit area is:
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33. With these approximations the cable equation becomes:
Linear cable equation
With these approximations the cable equation becomes:
Membrane time constant
Steady-state space constant
(This form of the cable equation assumes that the radii of the cable segments used to model a neuron are constant except at branches and abrupt junctions.)
In a simple form:
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34. Current injection at x=0:
Infinite cable
If the current injected into the cable is held constant, the membrane potential settles to a steady-state solution that is independent of time:
Where
Current injection at x=0:
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35. Infinite cable
The membrane potential produced by an instantaneous pulse of current injected at the point x=0 and the time t=0: 35

36. Voltage-dependent conductances
Single ion channels are either in an open or a closed state. The probability of the states can depend on the membrane potential and on the binding of various substances (e.g. neurotransmitters) to the cell membrane.
Figure: The currents passing through a single ion channel. In the open state, the channel passes -6.6pA at the holding potential of -140mV. This is equivalent to more than 107 charges per second passing through the channel.
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37. The Hodgkin-Huxley model
Conductance per unit area of membrane for channel type i:
Where
Pi is the probability of a single channel being in the open state. This is approximately equal to the fraction of open channels (if the number of channels is large).
is the maximal conductance per unit area of membrane.
Structurally, ion channel pores have several gates, which all need to be open for current to flow through the channel.
Example: In the case of the "delayed rectifier" K+ current:
This means for the K+ ions to pass, 4 gates need to be open.
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38. Gating equation
The transition of each gate is described by a kinetic scheme in which the gating transition closed-to-open occurs at voltage-dependent rate
and the reverse transition occurs at rate
n is the probability that we find a gate open.
Left: example transition functions plotted as a
function of the membrane potential
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39. Gating equation (other form)
The same equation in a useful form:
where
is called the time constant and
is the steady-state activation function.
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40. Simplified description: activation and inactivation gate(s)
Transient currents
Simplified description: activation and inactivation gate(s)
Example: transient Na+ current: The channel has k=3 activation and 1 inactivation gates:
activation – deactivation
deinactivation - inactivation
10/27/2010
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41. Na and K channel example
Left: steady-state activation (n,m) and inactivation (h) curves plotted for the Na and delayed rectifier K channel. It describes the fraction of channel gates that will be open if the membrane potential is clamped to V for a long time. For example the m gate will be ~50% open at -40mV.
Right: time constant functions for the gates. This function describes how fast the channel gate will reach its steady-state activation.
10/27/2010
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42. The full model of the giant squid axon with perfect „space clamp”:
Action potential
The full model of the giant squid axon with perfect „space clamp”:
The model includes:
Na conductance with 3 activation gates and 1 inactivation gate,
K conductance with 4 activation gates.
A passive leak conductance (the current flowing through the conductance is in linear relation with the membrane potential).
10/27/2010
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43. Figure B: Membrane potential during an action potential.
Squid axon HH model
Figure A: Currents flowing through the Na and K channels during an action potential.
Figure B: Membrane potential during an action potential.
Figure C: Activation and inactivation values during the spike.
10/27/2010
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44. Phenomenological models of synaptic conductances
Exponential (for example AMPA type glutamate receptor)
Difference of exponentials (for example GABAA). This method uses two time constants, thus both rise and decay can be described
Alpha function
This equation describes an isolated presynaptic release that occurs at t=0, reaches its maximum at the time constant, and decays with the time constant.
Figure: Example alpha function with Pmax=200 and tau=30.
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45. Causes of more complex behavior in real neurons
Lots of different types of ion channels
Complicated and cell-type dependent morphology
The temporal and spatial pattern of synaptic input
Neuromodulators, intracellular messenger molecules,...
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46. Biophysically detailed multicompartmental models
Biophysically detailed multicompartmental models
Figure: The electrical circuit representation of three compartments in a multicompartmental model. Circles with V represent voltage-gated conductances.
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47. Biophysically detailed multicompartmental models
The current flowing through compartment
, with neighbors
and
where
is the membrane potential of the compartment.
is the transmembrane current.
is the membrane capacitance of the compartment.
is the total injected electrode current.
is the total surface area of the compartment.
are the resistive couplings to the neighboring compartments.

48. Simulation software
For multicompartmental, detailed biophysical modeling:
Neuron (v. 7.4): flexible, reliable, powerful; also useful for detailed and more abstract network models -
GENESIS (v. 2.4 – old):
Moose (v. 3.1):
Neurospaces (GENESIS 3):
For efficient simulation of large-scale network models using simple model neurons:
NEST (v. 2.12):
Brian (v. 2.0):
For systematic model description, and simulation using external software (Neuron, GENESIS, NEST, etc.):
Neuroconstruct (v. 1.6):
PyNN (v. 0.8):
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49. Neuron
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50. Neuroconstruct
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51. Integrate-and-fire model
If the membrane potential reaches a threshold, the cell fires an action potential, and the membrane potential is set to a „reset potential”.
Simplest firing model: passive „integrate-and-fire”
Or:
Picture: Membrane potential trace of an integrate-and-fire model (Ie is the injected current).
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52. Integrate-and-fire model
In the case of constant current injection, the analytic solution:
Picture: integrate-and fire models compared to in vitro recordings.
Left: Firing rate as a function of the injected current. Continuous line: model results; filled circles: Results for the first two spikes fired, in vivo recordings; open circles: steady-state firing frequency, in vivo recordings.
Middle: In vivo recording from pyramidal cell.
Right: voltage trace of the adaptive integrate-and-fire model.
The adaptation of the firing rate and refractory states are relatively easy to implement.
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