Transcript Lecture notes

Modelling Gaseous
Neurotransmission
Computational Neuroscience
Lecture 10
“The nerve fibre is clearly a signalling mechanism of limited scope.
It can only transmit a succession of brief explosive waves, and the
message can only be varied by changes in the frequency and in the
total number of these waves. … But this limitation is really a small
matter, for in the body the nervous units do not act in isolation as
they do in our experiments. A sensory stimulus will usually affect a
number of receptor organs, and its result will depend on the
composite message in many nerve fibres.” Lord Adrian, Nobel
Acceptance Speech, 1932.
We now know it’s not quite that simple
• Single neurons are highly complex
electrochemical devices
• Synaptically connected networks are only
part of the story
• Many forms of interneuron communication
now known – acting over many different
spatial and temporal scales
Classical Neurotransmission
• Point-to-point transmission at synapses i.e. locally
• Occurs over a short temporal and spatial scale (2D)
• Overriding metaphor is electrical nodes connected by
wires
• Inspiration for standard connectionist ANN

Neuromodulation by gases
Recently neuromodulatory gases have been
discovered (NO, CO, H2S - all highly poisonous).
By far the most studied is NO
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Small and non-polar  freely diffusing
Act over a large spatial scale: volume signalling
Act over a wide range of temporal scales (ms to years)
4 dimensional diffusion
Modulatory effects
Interaction between neurons not connected synaptically
Loose coupling between the 2 signalling systems (electrical
and chemical) i.e. neurons that are connected electrically are
not necessarily affected by the gas and vice versa.
Neuromodulation by nitric oxide (NO)
• Implicated in learning and memory formation (esp. spatial)
and effects can be concentration dependent NO is different to
classical neurotransmitters
• No specific inactivating mechanism: NO is destroyed through
oxidation and
• Because of diffusive properties cannot be stored in vesicles
so whole neuron is potential release site
• Must be generated on demand and synthesis is coupled to
calcium influx and thus electrical activity
• highly reactive (and so poisonous) - used by white blood cells
for cell defence
• To understand NO’s action must therefore analyse its spatiotemporal spread
• But due to reactivity and diffusive properties very hard
to measure NO concentrations directly
• However, properties that make NO hard to measure
make it easy to model its diffusion mathematically
• Diffusion governed by free diffusion ie Fick’s 2nd law:
molecules diffuse from areas of high concentration to
low concentration at a rate proportional to the rate of
change of concentration gradient
C
2
 D C
t
D is the diffusion coefficient and measures the speed of diffusion (fast
for NO: 3300 cm2/s
Modified to incorporate NO destruction. No great knowledge of the
dynamics of destruction and so is generally modelled by first order
exponential decay via:
C
2
 D C   C
t
Where  is the decay rate so half-life = ln(2)/. In the brain general
background decay has a half-life of 5s.
However blood vessels etc represent highly oxidative local NO sinks
and have much shorter half life (< 1ms). Can incorporate these using
a spatially dependent decay rate (high inside sink and zero elsewhere
C
 D 2C  C   ( x)C
t
Can also add a production term to RHS though this can be factored
into equation via initial conditions
Methods
• 2 approaches to modelling NO diffusion
• In certain situations, equations can be solved
directly (analytical solution)
• However, solutions require numerical integration
and more complex source morphologies require
more numerical integration
• When this becomes prohibitive use difference
equation techniques to model spread
Previous models
• 2 styles of modelling used previously
• Compartmental models: very simple form of explicit (ie using
only values known at current timestep to estimate conc at next
timestep) finite difference
• OK for general conceptual work but hampered by ‘speed limit’ on
time/spatial scales that can be used (especially in higher
dimensions) where for stability:
Dt
1
(x)
2

2n
Where n is spatial dimension (timestep < diffusion time across a cell).
Leads to quite coarse approximations
Point source models
• Other style of modelling is to analytical model
• However, for simplicity it was assumed that NO from a cell can
be assumed to be created at an infinitesimal point at the centre of
the cell (cf gravitational approximation)
• This causes many problems as the source is inherently singular
and leads to unbiophysical results:
Concentration
during synthesis
at source is
infinite: cannot
assess the
internal
concentration
• Problems of singularity carried through to later timesteps
Lead to a large overestimation of the spatial extent of the signal
Also, ignores the role of the structure of the source and eg cannot
model hollow sources which are prevalent in the brain
Compartmental models also do not address role of morphology
Structure-Based Models
• To incorporate structure of source need to use (slightly) more
complex analytical model
• Method is to build up solution from point sources spread
throughout the source
EG can
examine the
solution for a
hollow
spherical
source
cytoplasm
synthesises NO
while nucleus
does not
See 2 novel features: reservoir of NO builds up in the centre (centre
effect). Reservoir is then ‘trapped’ until No in the cytoplasm has
dissipated
Because of this, the spread of NO is delayed leading to a delay in the
rise of No at distal points. Has signalling implications as there is then a
delay until it reaches effective concentrations
Can also examine tubular
sources where similar centre and
delay effects are seen
A natural focus of our analysis is to look at the extent of a volume
signal. Here we examine the effect of the size of the source on the
affected region. See firstly that there is an ‘optimal’ size and also that
sources < 5 micron in diameter have a very limited effect
See this more clearly here and notice that as small sources reach
steady-state quickly, affected region is not increased significantly by
increasing length of synthesis and sources < 3 micron have no effect
However, there are many instances in the brain
where there are sources of this small size. Here
we see the optic lobe of the locust where there are
ordered arrays of parallel NO expressing fibres
Mammalian
cortical plexus
• In the mammalian cerebral
cortex, while NO-expressing
cells are only 2% of cells, their
processes spread throughout
the cortex
• Known that NO from these
cells used in linking neuronal
activity to blood flow
• Majority of fibres are too
small (sub-micron) to generate
an effective NO signal
individually
For these neurons to have an effect they must combine their
production: get a new type of signalling where volumes of the brain are
targeted. What are the properties of NO signal from dispersed sources?
Do we still see centre effect? Yes, but spatial concentration profile is
flatter and more extensive
How about delay? Temporally, in conjunction with a threshold conc
flat profile to a delay not only at distal points but throughout the
source, and a very steep rise in the size of region affected
Also varying the spacing we see that this in turn leads to a greater
region over threshold and the possibility of a range of temporal
behaviour (could be interesting for signalling in ANNs …)
Also, we see that
this arrangement
has very good
properties in terms
of signalling in the
cortex and the finer
the fibres the better!
Delay can act as a low pass filter so only persistent activity increases
blood flow (some evidence for this), don’t get potentially dangerous
high concentrations, evens out some of randomness of growth process,
negates need to target blood vessels directly
Finally, see some evidence for this from more abstract ANN models
with diffusible modulator: shows potential for use of more abstract
models in teasing out putative functional roles for properties seen
Here we see how the gas
serves as a low pass
filter for visual input to a
robot in a shape
discrimination task
Similar to putative role
in the cerebral cortex