Dynamics of Neurons and Neuron Populations

1
Dynamics of Neurons and Neuron Populations

• by Walter J. Freeman,
• prepared by Marko Puljic

2
Neurons Make Up the Brains
schematic view of neuropil
schematic view of axon

• input dendrites and output through axon (lasts
  for a thousandth of a second).
• 2 main types
• projection neurons
• dendritic arbor to a diameter of up to a
  millimeter
• axon extends up to a meter
• local neurons (interneuron)
• dentritic arbor to a tenth of a millimeter (25-50
  times the diameter of the cell body)
• interneuron is like the local streets, whereas
  the projection neuron, is like a system of main
  roads.
• axon tips, synapses, are attached to the
  dendrites of other neurons.
• most projection neurons in the forebrain are
  excitatory, whereas interneurons are either
  excitatory or inhibitory.
• several thousand synapses on the dendritic tree
  of each neuron.
• competition for synaptic space
• inactive synapses decay and disappear, even the
  neuron may vanish
• lifelong growth and the maintenance of active
  connections provide the basis for leraning,
  remembering, and adapting through modifications
  of the numbers and strengths of synapses, (they
  require exercise).
• typically a million or more other neurons within
  the radius of the dendritic arbor of a given
  neuron
• each neuron connects with about 1 percent of the
  neurons within its reach (still at least ten
  thousand input and ten thousand output
  connections for each neuron)

3
Activities of a Neuron
dendrite W
axon P
P
trigger zone
synapse
P
W


Neuron
P
W
block
0
0
-
-


0
threshold
-
0
excitatory
inhibitory

• electrical potentials (energy used by a neuron)
  that a neuron generates across the neural
  membrane
• axon expresses its state in the frequency of its
  action potentials (pulse rate)
• energy is provided over entire length with a
  short delay
• one pulse at the time (axon needs recovery time)
• dendrite expresses its state in the intensity of
  its synaptic current (wave amplitude)
• integrate the pulse inputs dendritic wave is
  proportional to the total number of pulses
  dendrite receive (wave of current can be
  superposed on top of the currents form other
  synapses)
• dendritic current at a synapse rises rapidly
  during the thousandth of a second and than
  returns slowly
• neuron converts incoming pulses to waves, sums
  them, and transmits that train to all its axonal
  branches.
• outward-flowing current triggers zone more likely
  to fire below threshold pulse rate of neuron to
  increase
• Inhibitory synapse turns the current so it
  decreases the firing probability and the pulse
  rate of active neuron.

4
Microscopic vs. Mesoscopic
P
W


Ensemble
P0
P
W
-
-


-
rest
rest
excitatory
inhibitory

• single neuron is expressed with the flow of the
  loop current inside the neuron, (measured with an
  electrode inside the cell body)
• private, intracellular, microscopic view
• Microscopic pulse and wave state variables to
  describe the activity of the single neuron.
• time scale thousandths of a second and
  thousandths of a millimeter
• flow of the same current outside the neuron is
  also revealed by an electric potential, (same
  current, smaller amplitude, lower resistance)
• public, extracellular, mesoscopic view
• mesoscopic state variables to describe the
  collective activities
• time scale tenths of a second and tenths of a
  millimeter

5
Conversion Operation
dendrite W
axon P
P
trigger zone
synapse
P
W


Neuron
P
W
block
0
0
-
-


0
threshold
-
0
excitatory
inhibitory
P
W


Ensemble
P0
P
W
-
-


-
rest
rest
excitatory
inhibitory
6
Neural Connections
Connections apply to a neuron and to a masses of
neurons. Sensory neurons in the somatic,
auditory, gustatory, and olfactory systems
transmit in parallel with divergence, they dont
interact. Cortical neurons form neural
populations, they interact. indicates
excitation and indicates inhibition. Neurons
do not excite or inhibit themselves synaptically
because input from their own output is only one
among a million.
convergence
divergence
series
parallel
-
-


-

auto-feedback
cooperative-feedback (positive feedback)
negative-feedback
7
Mass Activity
P
W


Ensemble
P0
P
W
-
-


-
rest
rest
excitatory
inhibitory

• describe the mass activity in a local
  neighborhood by a pulse density
• recording from outside the cell simultaneous
  firing of the pulses of many neurons in a
  neighborhood.
• wave mode observe the amplitude of the wave
  density
• measuring the electrical potential difference
  between the surface and the depth of the cortex
  (outer layer of brain)
• population is a collection of local neighborhoods
  - cortical column
• wave-pulse conversion in the population has a
  sigmoid curve with limits.
• resting level of pulse activity is low but not
  zero
• neurons in population generate background
  activity by continually sending pulses to each
  other at random, whether or not there is sensory
  input or motor output?!

8
Sigmoid Curve Explanation for the Ensemble
P
W


Ensemble
P0
P
W
-
-


-
rest
rest
excitatory
inhibitory

• Wave-pulse
• as the wave density in a neighborhood goes to the
  inhibitory side
• pulse density goes to zero with decreasing firing
  probability of axons in neighborhood.
• as wave density goes to the excitatory side
• trigger zones in the population encounter the
  refractory periods progressively,
• pulse density approaches an upper limit, because
  neurons need to recover between pulses (as
  neurons in neighborhood are excited, there is an
  increase in number of cells still recovering from
  previous activity).
• Pulse-wave
• synapses cannot be driven too far outside their
  normal ranges
• wave-pulse precedes the pulse-wave and sets the
  boundaries

9
First Building Block of Neurodynamics

• mesoscopic state
• first step by which neurons collectively form
  activity patterns
• neurons cease to act individually
• activity level is determined by the population,
  not by the individuals.
• transformation of the neurons from one mode of
  existence to another is an example of the state
  transition.
• e.g.

• excitatory aggregate neurons below threshold
  excite each other in positive feedback
• neuron gives 100 pulses on average but receives
  only 80 in return, then those 80 pulses next give
  64, and so on through successive cycles until the
  activity returns to zero.
• ration of 0.8 is called the gain of the loop.

0.8

• when growth continues and each neuron receives
  120 pulses for each 100 it gives, gain is 1.2
• activity level can theoretically increase with
  each successive cycle around loop form 120 to 144
  and so on without the limit, but it doesnt
  happen because of saturation.
• individual refractory periods determine the upper
  limit of the sigmoid curve for the population.
• saturation reduces the gain, until the gain
  returns to unity and a nonzero steady state.

1.2

• excitatory population always comes to a steady
  level of activity, with no need of inhibition.
• population rebounds from inhibition or
  excitation, when perturbed, population is
  semiautonomous.

10
Mesoscopic Responses
no feedback
If the feedback gain is zero ( no feedback), the
impulse response decay quickly the form of the
postsynaptic potential of single neurons. With
positive feedback the response is prolonged. If
the gain is equal to one, the response to a pulse
lasts indefinitely. If the gain exceeds one, the
response increases until saturation. When
excitatory neurons (E) interact with inhibitory
(I) by negative feedback, the response
oscillates. The stronger the gain, the longer the
oscillation lasts. In cortex the ration of E to I
is 10 to 1. The return to a resting point, within
limits, reveals a point attractor, (return point
level regardless of intensity of the input).
Limits define the basin of the attractor. The
state transition form a point attractor at zero
activity to a nonzero point attractor gives
steady-state activity (first building block of
neurodynamics).

E
amplitude

time
positive feedback


E
E
amplitude


time
negative feedback


E
I
amplitude
-
time
11
Features of the Population
amplitude

• population has a point attractor
• population returns (is attracted) to the same
  level after it is stimulated

point attractor
time

• range of amplitudes defines state space of
  population

amplitude
time
amplitude

• population returns is the basin of attraction
• ball rolling to the lowest point of a bowl

basin
time
12
Basin of Attractions
basin of attractions in 3D (all points are
visited frequently)
13
Oscillations

Negative feedback between excitatory (E) neurons
and inhibitory (I) neurons produces
oscillation. Lower graphs show the state space
of an area of cortex with a plot of the
excitatory state variable on the horizontal axis
and the inhibitory state variable on the vertical
axis. Input shock rings at its characteristic
frequency until the ringing decays to the steady
state. The ringing in cortical activity is the
evoked potential. Oscillation through negative
feedback is the second building block of
neurodynamics. When the excitatory cells are
released form inhibition, they are free to
respond to the background activity, and so give a
new surge of excitation to the inhibitory
population. This starts another cycle of
oscillation with lower amplitude.
E
I
background
amplitude
-
c
a
a
b
d
zero
time
impulse


E
E
-
-


I
I
point attractor
-
-
a
b
excite I cells
excite E cells


E
E
-
-


I
I
a
-
-
c
d
inhibit I cells
inhibit E cells
14
Cycle of Oscillation
E
I

• when the excitatory cells are released form
  inhibition, they are again free to respond to the
  background activity, and so give a new surge of
  excitation to the inhibitory population.
• starts another cycle of oscillation, at lower
  amplitude, and it repeats until ringing dies
• frequency is somewhere between 20 and 100 cycles
  per second, gamma range
• decay rate is the rate of return to the basal
  level
• measure of ration on any peak to the preceding
  peak

• if the ratio and the gain exceeds unity
• state transition occurs because the population
  does not return to the point attractor
• oscillation grows until it encounters the
  nonlinear limitations, and there it stays.
• steady state oscillation, a limit cycle, is the
  third building block on neurodynmaics.
• oscillation is semiautonomous, self-sustaining
  and self-organized.
• stable additional excitatory or inhibitory input
  temporarily increase or decreased, but on release
  form input, the population returns to its basal
  oscillation

15
A State Transition
naive
attentive
A state transition is required to go from the
steady state of a point attractor to the
sustained oscillation of a limit cycle
attractor. The gain changes are due to synaptic
modifications with learning. When the oscillation
system is perturbed, it returns to the same
pattern of oscillation after further excitation
or inhibition over a wide range. That range
defines the basin of the attractor. The state
transition form a point attractor to a limit
cycle attractor is the third building block of
the self-organizing neurodynamics.
amplitude
time
habituated
after learning
before learning
I
I
naive
habituated
E
E
attentive
limit cycle attractor
point attractor
16
State Space of the Cortex
Schematic drawing of chaotic itinerancy.
Dynamical orbits are attracted to a certain
attractor ruin, but they leave via an unstable
manifold after a (short or long) stay around it
and move toward another attractor ruin. This
successive chaotic transition continues unless a
strong input is received.

• state space of the cortex comprise an attractor
  landscape with several adjoining basins of
  attraction, one for each class of learned stimuli
• activity of a sensory cortex is described with
  itinerant trajectory over its landscape
• there is a succession of momentary pauses in the
  basin of attractors to which the cortex travels
  once a learned stimulus has arrived
• attractors are shaped
• by the stimuli indirectly,
• by previous experience with those stimuli
• includes preafferent signals and neruomodulators
  as well as sensory input.
• input modifies the synaptic connectivity and
  thereby attractor landscape
• state transition arise activity of itinerant
  trajectories of brain
• governs experience as habitual behaviors
• landscape of attractors is responsible for
  reliable sequences of goal-directed behaviors