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Mathematics of Neural Systems

- Guttmans Phase Locking, Biological Rhythms,

Neural Networks Model, MATLAB Simulator,

Intercellular Communication - (Synaptic Transmission)

References

- Hoppensteadt and Peskin, Modeling and Simulation

in Medicine and the Life Sciences, 2nd Edition,

2002, Springer-Verlag New York, Inc., Chapter 6 - Keener and Sneyd, Mathematical Physiology,

Springer 1998 - Bio 301 Human Physiology Lecture Notes Neurons

and the Nervous System, www.biology.eku.edu/RITCHI

SO/301notes2.htm

Neural Systems

- Commonly modeled with electrical analog

representations. - Possible uses for neural system include
- 1) prosthetics
- 2) neurological disease treatment
- 3) pain management
- 4) control systems applications

Neural Systems

- Many differing models have been developed to

model the neural system. - A.L. Hodgkins (1948) application of a voltage

across a neural membrane yielded two

relationships - Class 1 Membrane voltage oscillates with

approximate constant amplitude, frequency

increases with increasing stimulus. - Class 2 Membrane voltage oscillates with a small

amplitude that increases with stimulus, frequency

remains approximately constant. - Both relationships occur in vivo
- Class 1 sensor neurons
- Class 2 motor neurons

Neural Systems

- Neurons voltage-controlled oscillators.
- Hoppensteadt-Peskin model the neuron as a

voltage-controlled oscillator, an integrated

circuit. - H-P discuss two neural systems
- Thalamus ability to sort signals and relay

message to neocortex - Hippocampus creation of theta-rhythm patterns,

based on inputs from the medial septum and the

enthorhinal cortex.

Considerations in Neuron Modeling

- Four neurophysiological considerations used in

neuron modeling - Frequency relations between inputs into a neuron

and its output - Inter-neuron communication is time dependent
- Inhibitory/Excitatory neuron connections
- Multiple time scale involvement in neuron firing

Phase Locking

- Phase locking is observed in both physical and

biological systems. - Observed as lyre strings or approximately same

physical dimensions where heard to vibrate at the

same frequency. - Case of oscillatory signals of the form
- cos(?tf)
- where t time msecs
- ? frequency, signal period 2p/ ?
- f phase deviation
- This relationship is described as FM radio

transmission and is described in H-P pg. 196

Phase Locking continued

- Phase locking roughly described as the ability to

achieve an output proportionally related to the

input. - Where output frequency response is a ratio of

the regular voltage oscillation frequency, ?, to

the stimuli frequency, µ. - Output frequency response ?/ µ
- 11 phase locking occurs when ?/ µ 1

Biological Rhythms

- Biological clocks move rapidly during certain

parts of the day and slower at later times. - The biological clock can be represented by a wall

clock with closely spaced time intervals around

the edge of a clock during morning times and

farther time interval spacing during the

afternoon.

Simple Clock Model

- Time minutes represented as radians around the

unit circle. - T ?t f
- Where T time
- t Greenwich mean time sec
- ? number of radians moved per second
- f longitude
- - Mathematical representation of a clock is

described by a differential equation - Simple clock model
- T ? (the rate of change of T is equal to ?)

Clock Modulating

- T phase variable
- ? frequency
- Coordinates of the tips of the hand, on the clock

are given by - xRcos(?) yRsin(?)
- where R length of the hand
- Note Counterclockwise rotation of the hand

unless ?lt0.

?

Clock Modulation

- Classic frictionless pendulum oscillations are

modeled by - ??2?0
- Solutions have the form ?Acos(?tf), where A and

f inititial condition constants - How is the harmonic oscillator related to the

simple clock? - Let ?Acos(?), where ? ?tf
- Then d?/dt ?
- Therefore, harmonic oscillator is related to a

simple clock by the rate change shown above,

provided that A?0.

Clock Modulating

- Simple clock is driven by a motor, which moves

time at a constant rate. However, the human

biological clock doesnt behavior in this manner. - Natural clocks can become modulated by many

things, for example solar time perception through

sensors. - Represented by ??f(2pt/1440)
- Where, fmodulating external signal with

period2p - ttime minutes, 1440min/day
- f passes through a full cycle when t1440.

Biological Clock Movement

- ??f(2pt/1440), describes an irregular movement

of the hand throughout the day. - Hand is accelerated when fgt0, decelerated when

flt0, due to time of the day or sunlight. - Recall that hand vector is represented by

(cos?,sin?). - Now ??tF(t)
- Where dF/dtf(2pt/1440)
- Gravity also contributes to the hand velocity as

when hand is traveling from top to bottom,

gravity contributes to a faster movement of the

hand, while the opposite is true when the hand is

traveling from bottom to top of the clock. - x??sin?
- Where ? measures time (?0 _at_ noon, ?p _at_ 6

oclock) - ?sin? is gravitys influence of the clock

hand - This is an example of feedback the rate which

the hand moves depends on its current position

Simple Clock vs. Biological Clock

- Simple clock has three parts
- Oscillating system (energy source)
- Pendulum
- Spring
- Electrical circuit
- Trigger mechanism
- Connects the energy source to the output
- Clock face
- Presents the output of the oscillator

- Biological clock
- Energy source
- Nerve cells metabolism
- Trigger mechanism
- Controlled by ionic channels in cell membrane
- Cell membrane capable of handling oscillations.
- Output
- Cycle of nerve voltage

Biological Clocks

- Time is not easily measured as the overall timing

of the clock differs depending on time of day,

light cycles and natural rhythms, known as

circadian rhythms which was first observed by

Aristotle. - Neurons are the timers in a biological clock.
- Biological clocks can modulate one another and

networks of neurons interact.

Voltage-Controlled Oscillators (VCO)

- Recall feedback example described earlier where

the rate at which the hand moves depends on its

current position. - A phase-locked loop (PLL) model, where 11 phase

locking occurs when ?/ µ 1. - PLL model described as ??cos(?)
- Where ? VCOs center frequency
- cos(?) VCOs output
- ??cos(?) represents the canonical model for

Hodgkins Class 1 neurons.

Excitatory and Inhibitory Relationships in Neural

Networks

- Two types of interrelationships between neurons
- Excitatory relationship
- N1 N2 , describes a network of two

neurons, neuron 1 (N1) having an excitatory

synapse impinging on neuron 2 (N2). - Where membrane potentials are modeled by cos?1

and cos?2, respectively. - ?1?cos?1, ?2 ?cos?2Acos(?1 f12)
- where Agt0, is the connection strength between N1

and N2. - f12 are time delays in the connection and those

due to external sources - Inhibitory relationship
- N1 -N2 , describes a network of two

neurons, neuron 1 (N1) having an inhibitory

synapse impinging on neuron 2 (N2). - Modeled as the excitatory except that Alt0.

- A network of M neurons is modeled as
- The factor Ck,j is the connection strength from

network element k to j, and fk,j is the phase

deviation due to the connection. These may be

constant or change with time. Ej describes an

external signal applied to the system.

Thalamocortical Circuit

- Two interacting neurons, one inhibitory and the

other excitatory, in which one fires a rapid

burst of action potentials followed by a slower

pulse from the other is a common in the brain. - This phenomena takes place in two time scales.
- Sensory input to the brain is processed in the

thalamus, which sorts and routes signals for

further processing in the entorhinal cortex and

the hippocampus. - The parallel structure circuit formed between in

excitatory and inhibitory cells is called the

reticular complex. - Thalamic cells fire rapidly while the reticular

cells fire (relatively) slowly. (Figure 6.2 of

H-P text illustrate this interaction)

Thalamocortical Circuit

- The thalamic cell (TC) interacts with the

reticular complex (RC) and eventually excites a

neocortical structure (NS). - The atoll model is used to model a single channel

from TC to NS - x5.0(1scosx-cosy) y0.04(1cosy10cosx)
- The ratio of time scales is 5/.04125, bursting

when sgt0.

Thalamocortical System

- Multiple parallel structures is the TH thru NC

single line connection described previously. TH

send an excitatory output to the RC and receives

an inhibitory signal from the RC, which then

routes the signal to NC. - Parallel structures connected by RC units and all

parallel connections are inhibitory.

Thelacortical System Network Model

- TH and RC cells
- xj5.0(1cosxj-cosyjsj(t))
- yj0.04(1cosyjtanh(2cosxj-10L(t)))
- NC projections
- Zj10(0.1coszj-cosyj)
- Where j0,,N and L accounts for the inhibition

between parallel structures, connected at the RC

cells. - Note a0.5(aa)

Where

ThalamocorticalSingle Channel Model

- We can see the single channel from stimulus to TH

to RC to NC, described from the previous

equations. - As mentioned earlier, there are two time scales

involved in the interaction between excitatory

and inhibitory cells in the brain. - These differing time scales are due to the rapid

firing, of a burst of action potential, followed

by the firing of a slower burst from the other

cell. - This phenomenon is shown as

ThalamocorticalSingle Channel Model (Atoll Model)

Thelacortical System Network Model

- There exists a chosen ratio of time scales

between the network constants of 1001. - The cos-functions mimic neuron action potentials,

while the tanh-functions is scales the inputs to

the RC to lie on the interval -1,1. Where the

positive terms are excitatory and negative terms

are inhibitory. - This model has been simulated by H-P and is shown

in Figure 6.5 in the text. This simulation

illustrates how the dominant frequency input to

the TH is the one which eventually dominates NC

activity and essentially suppresses the other

signals.

Simulation of the thalamocortical model

Simulation of the thalamocortical model

- Thalamocortical model based on 15 channels
- What does this figure represent?
- Top 15 lines coszj(t)
- Middle 15 lines cosyj(t)
- Bottom 15 lines cosxj(t)
- Stimulation applied at t0, 500, 1000.
- The largest frequency stimulus allows message

propagation to the neocortex. - Illustrates how the channel with the largest

frequency input to the TH, eventually dominates

NC activity as discussed earlier.

Hippocampus

- Responsible for possible roles in navigation and

memory. - Very complex brain structure.
- Hippocampus formation consists of a long axis,

with two smaller axis perpendicular to the

longer. Both smaller axis consists of three

regions, called the dentate gyrus. These are

known as the CA1 and CA3 field, both consisting

inhibitory and excitatory neurons. - Neurons operate with two frequencies
- Theta (?) frequency (5-12Hz)
- Gamma (G) frequency (40Hz)

Hippocampus

- Two inputs to the Hippocampus
- Entorhinal Cortex (EC)
- Medial Septum (MS)
- Slices of the hippocampus have been shown to

contain oscillations, without input, and that

these oscillations operate in the G frequency

range. - H-P model the thin slice oscillator, consisting

of inhibitory and excitatory cells, as a phase

locked loop, with a chain of oscillations along

the hippocampus long axis. (See Figure 6.6)

Hippocampus Oscillator Model

- Phases
- MS ?tfs
- EC ?tfc
- jth oscillator input phases
- ?s ?tf(N-j)? f
- ?c ?t?j? f
- Where ? ftime lag from signal propagation from

segment to segment.

Hippocampus Model Pattern Analysis

- Let ? segment oscillator phase,?s Input S

(Is) phase, ?C Input C (Ic) phase - Modeled as (T5Hz)
- Substituting for ?c and ?s yields

Substituting F?-Tt-f yields

Hippocampus Model Pattern Analysis

- Does this equation have a steady state (FF)?

If

Therefore, the oscillator will exhibit a theta

rhythm with the three parameters K, G-T, and ?

being the conditions for stability.

Hippocampus Spatiotemporal Pattern of Neural

Activity

- Taking a look at the actual pattern observed

along the long axis of the model for 32 segments,

we can see the output frequency vs. phase

deviation between their input signals.

Hippocampus Spatiotemporal Pattern of Neural

Activity

Intracellular Communication

- Many differing models of intracellular

communication. - Some important facts
- Occurs at synapse junction
- Synapse located between axon of one cell and

closely spaced, connected, dendrite of a second

cell.

Synapses

- Two types of synapses
- Electrical Synapse Typically muscle or cardiac

cells, connected through a gap junction in the

cell membrane, that form a relatively

nonselective, low resistance pore in which

electrical current or chemical species flow. - Chemical Synapse Neurons typically communicate

by the release of a chemical from one cell to

another.

Synapse

Chemical Synapse

- We will deal primarily with the chemical synapse

type as we are interested in neural intercellular

communication. - As mentioned earlier, the synapse is located at

the base of an axon of one neuron and the

dendrite, also known as the postsynaptic cell or

membrane, of the other neuron.

Chemical Neuron

- The very small region, 500 angstroms wide,

separating the two cells is called the synaptic

cleft. - As an action potential reaches the nerve

terminal, Ca2, voltage-gated, channels are

opened, allowing a sudden influx in the terminal. - This influx of Ca2 allows a neurotransmitter to

be released into the synaptic cleft, by

diffusion, which then binds to the postsynaptic

cell at receptors.

Chemical Synapses

- The binding of the neurotransmitter causes the

postsynaptic membrane potential to change. - The neurotransmitter is then removed from the

synaptic cleft by diffusion and hydrolysis.

Synapse

Chemical Synapses

- Due to close packaging of many neuron to neuron

connections, many earlier experiments carried out

on neuron to muscle cell connections. - Muscle cell response to neuronal stimulus is

termed an end-plate potential, or epp.

Epps

- Two types
- Minimum epp caused by low Ca2 concentrations

in response to an action potential, appear in

multiples of the spontaneous epp. - Spontaneous epp formed in the absence of

stimulation, caused by random background activity

in which vesicles fuse with the cell membrane

releasing ACh into the synaptic cleft.

Quantal Nature of Synaptic Transmission

- The neurotransmitter chemical, acetylcholine

(ACh) binds to ACh receptors, which in muscle

cells act as cation channels. - The flow of cation causes a change in membrane

potential. - It is now known that the packaging of ACh into

discrete vesicles results in quantal synaptic

transmission.

Simple Chemical Synapse Model

- Assumptions
- Synaptic terminal of the neuron consists of a

large number, n, of releasing units. - Each releasing unit releases a fixed amount of

ACh with a probability, p. - Release sites are independent, allowing a

binomially distributed number of quanta of ACh,

released by an action potential.

ACh Releasing Sites

- Probability that k sites fire

where firing refers to the release of a quantum

of ACh.

ACh Release Sites

- Under normal conditions, p is normally large.
- However, if there is some low extra cellular

concentration of Ca2, p will be small. - If n, which represents the total number of ACh

release sites, is large, and npm remains fixed,

where m is the mean or expected value, then the

binomial distribution assumption can be

approximated by a Poisson distribution.

ACh Release Sites

- Binomial distribution given by a Poisson

distribution

Mean Value, m

- Two was to estimate m.
- Method 1
- Notice that P(0) e-m.
- e-m number of action potentials with no epps
- total number of action potentials

Mean Value Calculation Method 2

- Due to our previous assumptions, a spontaneous

epp results from the release of a single quantum

of ACh, and a minimum epp is a linear sum of

spontaneous epps. - m mean amplitude of a miniature epp mean

amplitude of a spontaneous epp

Spontaneous Epps

- Dont have a constant amplitude, because amounts

of ACh released from each channel are not

identical. - Approximation, the amplitudes of single-unit

release, A1(x), normally distributed (Gaussian

distribution), with a mean µ and a variance a2. - Amplitude distribution can be calculated as

follows - If k vesicles released, the amplitude

distribution, Ak(x), will be normally distributed

with mean kµ and ka2.

Amplitude Distribution (Normally Distributed)

- Given by

Amplitude Distribution (Normally Distributed)

- There are clear peaks associated with a certain

number of quanta, yet these peaks are smeared out

and flattened by the normal distribution of

amplitudes. - This can be observed by an excerpt from Keener

and Snydes, Mathematical Physiology, Figure

7.5.

Amplitude Distribution (Normally Distributed)

Presynaptic Voltage-Gated Calcium Channels

- The process of chemical synaptic transmission

begins with an action potential reaching the

nerve terminal and opening voltage-gated Ca2

channels. - This causes a sudden influx of Ca2 which then

releases the neurotransmitter.

Chemical Synaptic Transmission

Presynaptic Voltage-Gated Calcium Channels

- Early experiments, with voltage clamp data, by

Llinas et al. (1976), on a giant squid synapse

yielded a model of the relationship between

current Ca2 and synaptic transmission. - Presynaptic voltage stepped up and clamping at

some constant level has shown that the

presynaptic Ca2 current Ica increases in a

sigmoidal fashion.

Presynaptic Voltage-Gated Calcium Channels

- How is this relationship modeled?
- First of all, let us assume that these channels

consists of n identical subunits. - Each of these subunits can be in one of two

states S (shut) and O (open). - Only when all n subunits are in state O can the

channel admit Ica.

Presynaptic Voltage-Gated Calcium Channels

- This process is modeled as

- Where the number of open channels is

proportional to On, where O is the number of open

channels

Presynaptic Voltage-Gated Calcium Channels

- The voltage dependence of the channels is given

by the opening and closing rate constants k1 and

k2.

constants

Where kBoltzmanns constant Tabsolute temp

Vmembrane potential z1 and z2 are the number of

charges moving across the width of the membrane

from shut to open and vice-versa, respectively.

Presynaptic Voltage-Gated Calcium Channels

Presynaptic Voltage-Gated Calcium Channels

- The unknown constants can be calculated by

fitting the voltage to the clamp data shown in

Figure 7.5

Where s0 total number of subunits

Presynaptic Voltage-Gated Calcium Channels

- Assuming that membrane potential jumps instantly

from 0 to V _at_ t0 and that o(0)0. - Then

Presynaptic Voltage-Gated Calcium Channels

- We can model the single-channel current as

Where ci and ce are the internal and external

Ca2 concentrations and PCa is the permeability

of the Ca2 channel.

Presynaptic Voltage-Gated Calcium Channels

- Knowing this we can not calculate the presynaptic

Ca2 current Ica.

Where

total number of channels

percentage of open channels

Presynaptic Voltage-Gated Calcium Channels

- From curve fitting Llinas was able to determine

some best-fit values for the 5 unknown constants.

These are

Shows charge dependence from conversion from

State S to O.

Shows no voltage dependence from conversion from

State O to S.

Fixed Parameters

Synaptic Suppression

- Steady-state percentage of open channels is

Synaptic Suppression

- Single channel current, j, is a decreasing bell

shaped function of V. - ICa is a bell shaped function of V as well, due

to it being the function of the product of V and

j. - Illustrated as Figure 7.6 in K-S.

Steady-state ICa

Synaptic Suppression

- As in our previous discussion with the

Thalamocortical model, this model also has two

time scales. - Fast time scale js dependence instantaneously

on the voltage - Slower time scale Voltage controls the number of

open channels

Synaptic Suppression

- What impact does the two differing time scales

have on the single channels? - A stepped-up voltage causes a decreasing in the

single channel current. - However, the number of open channels is low, so

the decrease in single channel current doesnt

greatly impact the overall Ica. - Looking at the longer time scale, slower time

scale, a greater number of channels start to

gradually open, due to the increase in voltage. - This response to a stepped-up voltage is shown in

K-S Figure 7.7

Synaptic Suppression

Synaptic Suppression

- A step-decrease in voltage causes an increase in

the single channel current. - However, the number of open channels is now high,

so the increase of single channel current greatly

increases Ica. - Looking at the longer time scale, the number of

channels start to gradually close, due to the

decrease in voltage, causing the overall current

to decrease slowly.

Synaptic Suppression

- Figure 7.7 shows three separate curves, a b and

c. These curves describe the following - A ICas response to a small positive steps

on/off switching is a monotonic increase followed

by a decrease. - B An increase in positive step to 70mV yields

the same monotonic increase, however the decrease

in preceded by a slight increase in current. - C A larger step of 150mV causes an initial

complete suppression, due to j0. When

suppression is released, a large voltage response

is seen, followed by a decrease to some resting

state. (Synaptic Suppression)

Real Model???

- Previous examples have been carried out under

clamped voltage conditions. This is not a

realistic approach as the action potential at a

nerve terminal is a time-varying voltage. - There also exists some debate as to the role Ca2

plays in neurotransmitter release. There are

actually two viewpoints on this issue - 1 Calcium Hypothesis Sudden influx of calcium

results in neurotransmitter release. (Modeled

earlier) - 2 CalciumVoltage Hypothesis Neurotransmitter

release is triggered by presynaptic membrane

potential, with Ca2 playing a regulatory role.

Conclusion

- There are many differing models tackling the

concept of synaptic transmission. - The model we have discussed makes many

assumptions that dont completely describe real

neuron behavior. - However, with the basic model we can see many

interesting relationships and characteristics of

neurotransmitter release, channel opening and

closing relationships to membrane voltage,

synaptic suppression and total channel current.