Non-local neuronal connectivity, wave control and migraine modelling

9 Jan 2009

Cripps Hall Library, University of Nottingham

Sponsored by the UK Mathematical Neuroscience Network.

Organisers: Markus Dahlem (Berlin) and Stephen Coombes (Nottingham)

This one-day workshop on large scale neuronal dynamics relating to migraine will cover a broad spectrum of topics including models of cortical dynamics, cortical spreading depression, reaction-diffusion and delayed integral models, functional and structural connectivity of the cortex and propagating wave analysis. The aim is to bring together experimentalists and theoreticians working on different aspects of migraine and to provide a forum for discussion about future modelling directions. The meeting will take place on Friday, January 9, 2009 in the Cripps Hall Library, University of Nottingham and is open to all. See below for travel information,

Registration is free and includes tea/coffe and a buffet lunch.

Please note that the number of participants will be limited to around 50 and that there will be no registration fee. Places will be assigned on a first come - first serve basis. Please send an email to Stephen Coombes - stephen.coombes@nottingham.ac.uk - for registration.

For those wishing to attend from within the UK some financial support is available to support travel costs. Priority will be given to PhD students and post-docs. Please contact Stephen Coombes directly to make a request for travel support by e-mailing stephen.coombes@nottingham.ac.uk.


Programme

Participants


Programme

9:00am - 10:30am Arrive Tea and Coffee
10:30am - 11:15am Markus Dahlem (Institut fuer Theoretische Physik, Berlin) Combining volume and synaptic transmission: a mathematical framework to investigate spreading depolarizations
11:15am - 12:00am Henry Tuckwell (MPI MIS, Leipzig)

Stochastic and geometrical effects on SD wave propagation

12:00am - 1:00pm Lunch Catered Buffet
1:00pm - 1:30pm Anthony Strong (Neurosurgery, King's Colleg London) Spatio-temporal properties and pathological effects of spreading depolarisations in human brain injury and its experimental models: a clinician's perspective
1:30pm - 2:15pm Wytse Wadman (Swammerdam Institute for Life Sciences, Amsterdam) The role of ionic changes in SD and seizure initiation studied with computer modelling
2:15pm - 3:00pm Oscar Herreras (Functional and systems neurobiology, Madrid) From single channels to the DC signal using a steady state model of Spreading Depression: get rigid to gain flexibility?
3:00pm - 3:30pm Break Tea and Coffee
3:30pm - 4:15pm Guillemette Chapuisat (LATP, Marseille) A phenomenological model of spreading depression
4:15pm - 5:00pm Axel Hutt (Laboratoire lorrain de recherche en informatique et ses applications, Villers-les-Nancy Cedex) Effects of finite axonal conduction speeds on the activity propagation in neural populations
5:00pm Stephen Coombes and Markus Dahlem Concluding Remarks followed by trip to Amores restaurant for food and drinks (unfunded!)

Markus Dahlem

Combining volume and synaptic transmission: a mathematical framework to investigate spreading depolarizations

I will present spreading depolarization (SD), that is, a wave phenomena occurring in migraine, from the perspective of nonlinear dynamics, in particular from theories of spatio-temporal pattern formation. My main goal is to establish links between an abstract mathematical description and experimental data obtained by several techniques such as laser speckle imaging, direct electrocorticographic recordings,  high-field functional magnetic resonance imaging, and patient symptoms reports.  As is reflected by the selected choice of experimental findings, the spatio-temporal patterns in SD can extend over several centimeters.  Accordingly, the mathematical description considers large-scale neuronal activity  in populations of neurons rather than detailed models of single-ion channels and synapses. However, other talks by Wadman and Herreras in this workshop will focus on these aspects and it will be a challenging task, taken by Tuckwell, to bridge this gap.  The spatio-temporal pattern occur also on rather long time scales of up to half an hour. Space and time scales indicate that diffusion is the means by which these patterns are formed in the cortex, a concept called volume transmission.  The basic mathematical equation will therefore be a parabolic partial differential equation describing the reaction-diffusion (RD) system.  I will show that already a generic RD system that exhibits pulse solutions is able to reproduce most features of the spatial-temporal pattern observed experimentally. Some of these findings are known for more than four decades, but there is a renewed interest in  this approach now that recent experiments provide quantitative data of SD in human.  A particular solution of the RD system that matches the experimental findings best is, however, unstable.  We show that this solution becomes stable, when the RD model is augment with integro-differential terms that can represent cortical functional and structural connectivity.  Modeling cortical activity through local approximations of integral neural field equations will be a particular focus of the talk by Hutt.


Henry Tuckwell

Stochastic and geometrical effects on SD wave propagation

Spreading depression involves an enormous number of physiological variables with origins in glia, neurons and in extracellular space and whose dynamics are intertwined.   The reaction-diffusion models which focus on ionic and transmitter concentrations and movements have proven a useful first approach, especially with the recent development of models in 2 space dimensions (see AIP Conf. Proc. 1028 46 (2008)). Stochastic effects in 2 space dimensions will be considered with a view to estimating the likelihood that SD could "spontaneously' develop either through intense ("normal") local neural activity or epilepsy. Geometrical and physiological factors involved in reverberating waves and spiral waves will also be discussed.

Relevant publications

H C Tuckwell and R M Miura 1978 A mathematical model for cortical spreading depression, Biophysical Journal, Vol 23, 258-276.
H C Tuckwell 1980 Predictions and properties of a model of potassium and calcium ion movements during spreading cortical depression, International Journal of Neuroscience, Vol 10, 145-164.
H C Tuckwell 1981 Simplified reaction-diffusion equations for potassium and calcium ion concentrations during spreading cortical depression , International Journal of Neuroscience, Vol 12, 95-107.
H C Tuckwell 1981 Ion and transmitter movements during spreading cortical depression, International Journal of Neuroscience, Vol 12, 109-135.
H C Tuckwell 2008 Mathematical modeling of spreading cortical depression: spiral and reverberating waves, American Institute of Physics Conf. Proc., Vol 1028, 46-64.

Anthony Strong

Spatio-temporal properties and pathological effects of spreading depolarisations in human brain injury and its experimental models: a clinician's perspective

Experimental models of focal ischaemic stroke, principally those based on occlusion of the middle cerebral artery (non-recovery), have demonstrated that spontaneous depolarisations similar to spreading depression occur and spread in ischaemic boundary zones ("the penumbra"), and that ultimate infarct size depends on number of depolarisations; in the context of brain ischaemia new serial methods of mapping cortical perfusion have shown that these so-called peri-infarct depolarisations can exert a profound vasoconstrictor effect on the cerebral microcirculation- the opposite of the response in normally perfused brain. Collaborative work by the COSBID group (www.cosbid.org) monitoring depolarisations in patients with ischaemic or traumatic brain injury, together with further work in the laboratory has identified several factors that influence frequency of depolarisations and hence the extent and severity of damage. The experimental methods now available offer options to test and refine existing and novel models of depolarisation behaviour.


Wytse Wadman

The role of ionic changes in SD and seizure initiation studied with computer modelling

Computer modelling was used to gain insight in the initiation of spreading depression and seizures in a relatively simple model that comprises neuron-glia interactions. Although the limitations of such a model are evident (we did not extend it to include the spreading part, other speakers will deal with that in great detail), it nevertheless helped enormously to understand the very complex and intricate control systems in the brain that involve ionic homeostasis. We constructed neurons with limited and also with extended dendritic morphology and with a variety of membrane ionic channels (potassium, sodium, chloride and calcium) to provide realistic firing patterns, although the full realm of detailed properties at the millisecond scale was not covered. Ionic fluxes that carried the currents were incorporated as well as active pumps that controlled homeostasis in the intracellular compartments (glia and neuron) and the interstitial extracellular space between the neurons and glia. Changes in ionic strengths lead to swelling of the cells and changes in relative size of the volume compartments and these were included. The potassium buffering function of the glia syncitium was also implemented and studied. It is clear that models of this complexity can never prove an hypothesis, but they are extremely useful in understanding the complex behaviour of biological systems. Surprisingly, no additional features are necessary in our model to grasp the essential properties of SD and seizure initiation as experimentally observed.


Oscar Herreras

From single channels to the DC signal using a steady state model of Spreading Depression: get rigid to gain flexibility?

We combine core-conductor and field theories to explore the mechanism of generation of the negative DC potential (Vo) associated to SD. According to the former, if neurons depolarize completely they cannot produce transmembrane currents and hence contribute to the negative Vo. A way out for neurons may be in our intradendritic recordings, which revealed that depolarization and membrane shunt affect specifically certain zones of the dendritic anatomy. By computer-assisted reproduction of the extracellular potentials generated by model neurons in a simulated CA1 field under SD conditions we examined the contribution of different membrane channels to the Vo negativity using a steady-state approach. The model confirmed that individual neurons may support standing internal gradients of depolarization, which are necessary to rise Vo shifts. It also generated other predictions (tested in part): 1.- Standard V-dependent channels and Glu receptors play only a subsidiary role on Vo production (another membrane conductance is required). 2.- Spatial cancellation of transmembrane currents in single cells settles the amplitude and laminar distribution of the Vo in the hippocampus. The proposed mechanism for Vo production is, as first proposed by Leão himself, based on differential polarization of neuron membranes, just as for regular synaptic or spike potentials. We’ll discuss the peculiarities of the SD case and their implications.


Guillemette Chapuisat

A phenomenological model of spreading depression

Spreading depressions are depolarization waves that propagate in the gray matter of the brain. They are strongly suspected to be responsible for the aura during migraine. They also have a great importance in stroke or epilepsy. In this talk, I will present a phenomenological model of spreading depression composed of a system of 3 reaction-diffusion equations coupled with 2 ordinary differential equations. Using this model, I will study why spreading depressions very often stop in the Rolando sulkus and why spreading depressions are so difficult to observe in the human brain.


Axel Hutt

Effects of finite axonal conduction speeds on the activity propagation in neural populations

The propagation speed of action potentials along axons is finite and depends on various axonal physiological properties, such as diameter or the degree of myelination. Since the physiological properties of axonal trees may differ from brain area to bran area and the axonal tree itself does exhibit a spatially heterogeneous distribution of diameters and degrees of myelination, the axonal conduction speed obeys a statistical distribution. The presented work shows the mathematical analysis of a neural field
model involving distributed axonal conduction speeds. The work discusses the cases of spatially diffusive and nonlocal interactions and explains the analytical calculations of wave instabilities and traveling fronts.


Participants

Azad Abul Mathematics Research Institute, Exeter
Peter Ashwin Mathematics Research Institute, Exeter
Basabdatta Sen Bhattacharya Computer Science, Manchester
Nicholas Blockley Sir Peter Mansfield Magnetic Resonance Centre, Nottingham
Guillemette Chapuisat CMLA, Cachan Cedex
Stephen Coombes School of Mathematical Sciences, Nottingham
Jonathan Crofts Mathematics, University of Strathclyde
Markus Dahlem Institut fuer Theoretische Physik, Berlin
Tony Gardner-Medwin Physiology, UCL
Ana Maria Gheorghe School of Mathematical Sciences, Nottingham
Alvaro Guevara Maths, Louisiana State University
Oscar Herreras Functional and systems neurobiology, Madrid
Axel Hutt Laboratoire lorrain de recherche en informatique et ses applications, Villers-les-Nancy Cedex
Azadeh Khajeh-Alijani Mathematics Institute, University of Warwick
Peter Liddle Psychiatry, University of Nottingham
Robert Mackay Mathematics Institute, University of Warwick
Paul Matthews School of Mathematical Sciences, Nottingham
David Ray Biomedical Sciences, University of Nottingham
Hannelore Rittmann-Frank Institut fuer Theoretische Physik, Berlin
Aman Saleem Dept of Bioengineering, Imperial College London
Simon Schultz Dept of Bioengineering, Imperial College London
Helmut Schmidt School of Mathematical Sciences, Nottingham
Felix Schneider Institut fuer Theoretische Physik, Berlin
Anthony Strong Neurosurgery, King's College London
Carl-Magnus Svensson School of Mathematical Sciences, Nottingham
Ioannis Taxidis School of Mathematical Sciences, Nottingham
Christopher Tench Division of clinical Neurology, Nottingham
Ruediger Thul School of Mathematical Sciences, Nottingham
Yulia Timofeeva Department of Computer Science and Centre for Complexity Science, Warwick
Henry Tuckwell MPI MIS, Leipzig
Hao Zhu School of Mathematical Sciences, Nottingham
Wytse Wadman Swammerdam Institute for Life Sciences, Amsterdam
Kyle Wedgwood School of Mathematical Sciences, Nottingham


Directions to Nottingham

Airport links

International airports in order of nearness to Nottingham:

East Midlands Airport has scheduled services to several UK and European airports. There is a bus service to Nottingham city centre: journey time is about 30 minutes.

From East Midlands Airport you can take the Trent Barton Rainbow 5 service. Buses leave from outside of the Airport Arrivals hall. You can also walk to the taxi rank on the terminal forecourt and take a direct taxi to the University. The cost of a single/one way journey is approximately £25. Taxis are normally available 24 hours. From M1 Motorway: Leave the M1 motorway at Junction 25 to join the A52 to Nottingham. Turn right at The Priory roundabout (about 4 miles from M1), then left at next roundabout to enter the University's West Entrance.

Birmingham Airport. There are frequent trains from the airport station (Birmingham International) to Birmingham New Street, from where there are regular trains to Nottingham. The journey takes up to 100 minutes.
Manchester Airport. There are frequent trains from Manchester Airport Station to Manchester Piccadilly Station, from where there are regular trains to Nottingham. The journey takes around two hours.
London Heathrow Airport has no surface rail link direct to Nottingham. You can take the underground to King's Cross - St Pancras mainline station, journey time is about 70 minutes. Alternatively, you can take the expensive Heathrow Express (journey time 15 minutes) or the less expensive Heathrow Connect (journey time 30 minutes) to Paddington Station. From there, go to St. Pancreas from where there are frequent trains to Nottingham. The journey by train takes almost 100-120 minutes.
London Gatwick Airport. By train ignore the Inter-City Gatwick Express to London Victoria. Instead take the Thameslink service to either King's Cross Thameslink (200m from St. Pancras) or Luton or Bedford, both on the London--Nottingham line. Enquire locally which change of train is most appropriate

Railway links

Nottingham station. From there one can reach the campus by taxi (about 15 minutes), or by bus. Many busses go to the QMC (Queen's Medical Centre) which is close to the campus.
Beeston station, about five minutes before Nottingham. From here it is only a 40 minute walk to the venue.

The Trent Barton Rainbow 5 service runs from Beeston Bus Station (about 10 minutes walk from the railway station along Station Road) past the South and North entrances to the University, as well as passing QMC. Nottingham City Transport service 13 has a stop in Queen's Road, about 5 minutes walk from the Beeston railway station, turning right out of Station Road.

Local directions to Cripps Hall Library

Map showing the location of the University of Nottingham and the University Park campus
Cripps Hall is close to Building 19 (Cripps Health Centre) on the campus map.

Further travel information can be found here.


S Coombes | Center for Mathematical Medicine | School of Mathematical Sciences | University of Nottingham