Computational Properties of Extracellular Calcium Dynamics in Mammalian Neural Tissue

David M. Eagleman and P. Read Montague 

One fifth of the mammalian brain is extracellular space (ECS). The extracellular space is not empty, but instead comprises a complex network of proteins and a variety of molecular species. One extracellular species, calcium, holds a prominent position as one of the most important messengers known in the brain. However, calcium exists in low concentrations in the ECS, and its diffusion is slowed by the restricted volumes of the extracellular space - therefore, normal neural activity may cause calcium to flow out of the ECS faster than diffusion can fill it in. As opposed to the traditional view that extracellular calcium exists at a stable concentration, we explore the hypothesis that calcium concentrations may change. Such changes would be expected to carry significant functional impact, due to the many roles calcium plays. The hypothesis is explored both at the biophysical and theoretical levels. At the biophysical level, we explore the dynamics of external calcium changes under a wide range of reasonable parameter settings. At the theoretical level, we attempt to interpret how such fluctuations may serve as information-bearing signals in the nervous system.

Specifically, we have employed Monte Carlo simulations, finite differencing schemes, and numerical analyses to develop a computational method capable of describing populations of neural elements, and the fluids that communicate through the spaces between them. Such a method has allowed me to address several issues, the broadest interpretations of which can be cast as a set of questions: Is neural tissue engineered to allow - or prevent - fluctuations in extracellular ionic concentrations? If fluctuations can occur, do they carry information? What are the computations carried out by that flow of information?

Beginning with a model of pre-synaptic terminals, we show that reasonable assumptions about the kinetics of consumption and diffusion will lead to rapid, local changes in external calcium. The exact size of the calcium signal depends on several parameters. For example, the cleft width might be used by the tissue as a control parameter: changing the gap between elements can amplify or squelch the calcium signal. The distribution of calcium channels will also critically influence the calcium fluctuations: by clustering the channels, the local signal amplitude is greatly increased. In some circumstances, the density of the calcium channels can become high enough that the total calcium influx becomes limited by the speed of extracellular diffusion. We calculate that the calcium signal will not travel far through the tissue - the signal will remain approximately as local as neurotransmitter signals such as glutamate. Finally, we show that changes in the average background rates will significantly change the average calcium concentration available to any given terminal.

Moving the focus to dendrites, we demonstrate that action potentials propagating along a dendrite can induce large calcium fluctuations, lowering the external calcium available to overlying pre-synaptic terminals. Since neurotransmission depends on the availability of external calcium, it may be that a post-synaptic neuron can employ back-propagating action potentials to modulate the transmission probabilities of overlying afferent terminals. The geometrical distribution of calcium sinks again influences the time and spatial extent of fluctuations in external calcium. In particular, clusters of co-active dendrites can prolong and amplify an external calcium fluctuation. This latter effect provides a natural substrate for a computational mechanism that indexes (locates) specific volumes of neural tissue on rapid time scales.

We suggest that the detailed structure of the extracellular space, in combination with the three-dimensional distribution of calcium sinks, will play a role in neural information processing. We discuss many roles extracellular calcium is known to serve - however, throughout this work, we highlight two roles that are directly interpretable from a computational perspective: neurotransmission and tissue re-organization. The steep dependence of neurotransmitter release on calcium allows for interpretation of calcium dynamics at short time scales. Over longer epochs, we appeal to the calcium-dependence of certain call-adhesion molecules to propose that calcium dynamics may drive learning in neural tissue. Both explorations will underscore the dual role that calcium plays: when an ion moves from outside to inside, its presence in the cytoplasm is as important as its absence from the ECS.

In conclusion, the geometrical arrangement of neural tissue forces neighboring neural elements to share a resource that is necessary for neural transmission, but is in limited supply on short temporal and spatial scales. As we demonstrate, the resting levels of external calcium are not sufficiently high to protect against large decrements in this important resource. Instead, it seems as though the tissue is engineered so that external calcium levels are meant to fluctuate dramatically; given the functional importance of external calcium, we are led to the strong suspicion that external calcium fluctuations are an important class of information-bearing signal in the nervous system.

 Related Publications:

Wiest, M.C., Eagleman, D.M., King, R.D., Montague, P.R. (2000) Dendritic spikes and their influence on extracellular calcium signaling. Journal of Neurophys iology. 83(3):1329-1337.

King, R.D., Wiest, M.C., Montague, P.R., Eagleman, D.M. (2000) Do extracellular calcium signals carry information through neural tissue? Trends in Neuroscie nces 23(1):12-13. 

Eagleman, D.M., Montague, P.R. (1999) Calcium dynamics in the extracellular space of mammalian neural tissue. Biophysical Journal 76(4):1856-1867. 

Eagleman, D.M. (1998). Doctoral dissertation: Computational properties of extracellular calcium dynamics. 

Eagleman, D.M., Montague, P.R. (1998). Computational properties of peri-dendritic calcium fluctuations. Journal of Neuroscience. 18(21): 8580-8589. 

Eagleman, D.M., King, R.D., Montague, P.R. (1998). Interaction of nitric oxide and external calcium fluctuations: a possible substrate for rapid information retrieval. Prog Brain Res 1998;118:199-211.
(Also pubished in Nitric Oxide and other diffusible messengers in development, plasticity, and disease. Mize,R.R., Friedlander,M.J., Dawson,T.J., Dawson,V.M., Eds. Amsterdam: Elsevier Press.)

Eagleman, D.M., Wiest, M.C., Montague, P.R. (1998). Extracellular calcium dynamics as a rapid information-bearing signal in neural tissue. Soc. Neurosci. Abstr. 24:526.

Montague, P.R. (1996) The Resource Consumption Principle: attention and memory in volumes of neural tissue. Proc. Nat. Acad. Sci. USA 93(8):3619-3623. 

Montague, P.R. (1996). General properties of the resource consumption principle of neural function. J Physiol Paris, 90(3-4):239-42.

Person, C., Eagleman, D.M., King R.D., Montague PR. (1996) Three-dimensional synaptic distributions influence neural processing through the resource consumption principle. Journal of Physiology, Paris, 90(5-6): 323-5.