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.