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Mathematics of Neural Systems

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Title: Mathematics of Neural Systems


1
Mathematics of Neural Systems
  • Guttmans Phase Locking, Biological Rhythms,
    Neural Networks Model, MATLAB Simulator,
    Intercellular Communication
  • (Synaptic Transmission)

2
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

3
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

4
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

5
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.

6
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

7
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

8
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

9
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.

10
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 ?)

11
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.

?
12
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.

13
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.

14
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

15
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

16
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.

17
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.

18
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.

19
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)

20
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.

21
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.

22
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
23
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

24
ThalamocorticalSingle Channel Model (Atoll Model)
25
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.

26
Simulation of the thalamocortical model
27
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.

28
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)

29
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)

30
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.

31
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
32
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.
33
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.

34
Hippocampus Spatiotemporal Pattern of Neural
Activity
35
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.

36
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.

37
Synapse
38
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.

39
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.

40
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.

41
Synapse
42
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.

43
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.

44
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.

45
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.

46
ACh Releasing Sites
  • Probability that k sites fire

where firing refers to the release of a quantum
of ACh.
47
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.

48
ACh Release Sites
  • Binomial distribution given by a Poisson
    distribution

49
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

50
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

51
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.

52
Amplitude Distribution (Normally Distributed)
  • Given by

53
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.

54
Amplitude Distribution (Normally Distributed)
55
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.

56
Chemical Synaptic Transmission
57
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.

58
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.

59
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

60
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.
61
Presynaptic Voltage-Gated Calcium Channels
62
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
63
Presynaptic Voltage-Gated Calcium Channels
  • Assuming that membrane potential jumps instantly
    from 0 to V _at_ t0 and that o(0)0.
  • Then

64
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.
65
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
66
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
67
Synaptic Suppression
  • Steady-state percentage of open channels is

68
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.

69
Steady-state ICa
70
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

71
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

72
Synaptic Suppression
73
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.

74
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)

75
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.

76
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.
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