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 gasesRecently neuromodulatory gases have been discovered (NO,

CO, H2S - all highly poisonous). By far the most studied is NO

• 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 spatio-temporal 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

CDt

C 2

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:

CCDt

C 2

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

CxCCDt

C)(2

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:

nx

tD

2

1

)( 2

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 …)

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

Also, we see that this arrangement has very good properties in terms of signalling in the cortex and the finer the fibres the better!

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