The area of neural simulation began with the classic Hodgkin and Huxley model of the action potential (Hodgkin and Huxley 1952). At that point calculating a single action potential using a manually cranked calculator took 8 hours of hard manual work. Since then the ability to compute neural activity across large networks has developed enormously thanks to increases in computer power.
http://neuron.duke.edu/userman/2/pioneer.html
What is needed?
What information does it need for a given resolution?
It is known that the morphology of neurons affects their spiking behaviour (Ascoli 1999), which suggests that neurons cannot simply be simulated as featureless cell bodies. In some cases simplifications of morphology can be done based on electrical properties (REF: Rall etc).
One of the most important realisations of recent computational neuroscience in recent years is that neurons in themselves hold significant computational resources. “Dendritic computing” involves nonlinear interactions in the dendritic tree, allowing parts of neurons to act as ANNs on their own (Single and Borst 1998; London and Hausser 2005; Sidiropoulou, Pissadaki and Poirazi 2006). It appears possible that dendritic computation is a significant function that cannot be reduced into a whole-cell model but requires calculation of at least some neuron subsystems.
Brain emulation need to take chemistry more into account than commonly occurs in current computational models (Thagard 2002). Chemical processes inside neurons have computational power on their own and occur on a vast range of timescales (from sub-millisecond to weeks). Neuromodulators and hormones can change the causal structure of neural networks
About 200 chemical species have been identified as involved in synaptic plasticity, forming a complex chemical network. However, much of the complexity may be redundant parallel implementations of a few core functions such as induction, pattern selectivity, expression of change, and maintenance of change (where the redundancy improves robustness and the possibility of fine-tuning) (Ajay and Bhalla 2006).
Proteomics methods are being applied to synapses, potentially identifying all present proteins (Li 2007).
http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pubmed&pubmedid=17098931
synapse protein database
At the very low numbers of molecules found in synaptic spines chemical noise becomes a significant factor, making chemical networks that are bistable at larger volumes unstable below the femtoliter level and reducing pattern selection (Bhalla 2004b; Bhalla 2004a). It is likely that complex formation or activity constrained by membranes is essential for the reliability of synapses.
In many species there exist identifiable neurons, neurons that can be distinguished from other neurons in the same animal and identified across individuals, and sets of equivalent cells that are mutually indistinguishable (but may have different receptive fields) (Bullock 2000). While relatively common in small and simple animals, identifiable neurons appear to be a minority in larger brains. Early animal brain emulations may make use of the equivalence by using data from several individuals, but as the brains become larger it is likely that all neurons have to be treated as individual and unique.
Review of models signalling
http://www3.interscience.wiley.com/cgi-bin/fulltext/109659397/PDFSTART
model collections
http://senselab.med.yale.edu/BrainPharm/NeuronDB/ndbNeuronList.asp
http://senselab.med.yale.edu/senselab/NeuronDB/default.asp
http://www.ebi.ac.uk/compneur-srv/LGICdb/LGICdb.php
http://www.iuphar-db.org/iuphar-ic/index.html
http://www.ionchannels.org/
http://www.nimh.nih.gov/neuroinformatics/index.cfm
http://citeseer.ist.psu.edu/symanzik99visual.html
Visual Data Mining of Brain Cs
Visual Data Mining of Brain Cells
(effect of morphology on functional properties)
An issue that has been debated extensively is the nature of neural coding and especially whether neurons mainly make use of a rate code (where firing frequency contains the signal) or the exact timing of spikes matter (Rieke et al. 1996). While rate codes transmitting information have been observed there exist fast cognitive processes (such as visual recognition) that occur on timescales shorter than the necessary temporal averaging for rate codes, and neural recordings have demonstrated both precise temporal correlations between neurons (Lestienne 1996) and stimulus-dependent synchronization (Gray et al. 1989). At present the evidence that spike timing is essential is incomplete, but there does not appear to be any shortage of known neurophysiological phenomena that could be sensitive to it. In particular, spike timing dependent plasticity (STDP) allows synaptic connections to be strengthened or weakened depending on the exact order of spikes with a precision <5 ms (Markram et al. 1997; Bi and Poo 1998). Hence it is probably conservative to assume that brain emulation needs at time resolution smaller than 0.4–1.4 ms (Lestienne 1996) in order to fully capture spike timing.
Neural Models
The first neural model was the McCulloch-Pitts neuron, essentially binary units summing weighted inputs and firing (i.e. sending 1 rather than 0 as output) if the sum was larger than a threshold (McCulloch and Pitts 1943; Hayman 1999). This model and its successors form the basis of most artificial neural network models. They do not have any internal state except the firing level. Their link to real biology is somewhat tenuous, although as an abstraction they have been very fruitful.
More realistic models such as “integrate-and-fire” sum synaptic potentials and produce spikes.
Conductance-based models are the simplest biophysical representation of neurons, representing the cell membrane as a capacitor and the different ion channels as (variable) resistances. Neurons or parts of neurons are replaced by their equivalent circuits, which are then simulated using ordinary differential equations. Beside the membrane potential they have (at least) two gating variables for each membrane current as dynamical variables.
The core assumptions in conductance-based models is that different ion channels are independent of each other, the gating variables are independent of each other, depending only on voltage (or other factors such as calcium), first order kinetics in the gating variables and that the region being simulated is isopotential.
More complex ion channel models with internal states in the channels have been developed, as well as models including calcium dynamics (possibly with several forms of calcium buffering)
http://icwww.epfl.ch/~gerstner/SPNM/node11.html
Synapses can be modeled using exponentials, beta functions
Destexhe 1994
Learning rules, synaptic adaptation
http://icwww.epfl.ch/~gerstner/SPNM/node69.html
In simple neuron models the “neuronic” (firing rate update) equations can be uncoupled from the “mnemonic” (synaptic weight update) equations, the “adiabatic learning hypothesis” (Caianiello 1961).
However, realistic models often include a complex interplay at synapses between membrane potential, calcium levels and conductances that make this uncoupling hard.
Marom S and Abbott LF. "Modeling state-dependent inactivation of membrane currents." Biophys J. 1994 Aug;67(2):515-20.
Parameters used in conductance-based models are today derived using voltage-clamp experimental data
Willms AR. "NEUROFIT: software for fitting Hodgkin-Huxley models to voltage-clamp data." J Neurosci Meth. 2002, 121:139-150.
Reducing models to simpler ones
(Izhikevich 2004) reviews both typical neural firing patterns and a number of computational models of spiking neurons.
(from (Izhikevich 2004), figure 2)
He estimates both the number of possible biological features different models can achieve and how many floating point instructions are needed per ms of simulation (only assuming a soma current, not taking the effects of dendrites and synapses into account):
Model # of biological features FLOPS/ms
Integrate-and-fire 3 5
Integrate-and-fire with adapt. 5 10
Integrate-and-fire-or-burst 10 13
Resonate-and-fire 12 10
Quadratic integrate-and-fire 6 7
Izikhevich (2003) 21 13
FitzHugh-Nagumo 11 72
Hindmarsh-Rose 18 120
Morris-Lecar 14* 600
Wilson 15 180
Hodgkin-Huxley 19* 1200
* Only the Morris-Lecar and Hodgkin-Huxley models are “biophysically meaningful” in the sense that they attempt actually to model real biophysics, the others only aim for a correct phenomenology of spiking.
The (Izhikevich 2003) model is interesting since it demonstrates that it may be possible to improve the efficiency of calculations significantly (two orders of magnitude) without losing too many features of the neuron activity. The model itself is a two-variable dynamical system with two model parameters. It was derived from the Hodgkin-Huxley equations using a bifurcation analysis methodology keeping the geometry of phase-space intact (Izhikevich 2007). While it is not directly biophysically meaningful it, or similar reduced models of full biophysics, may be possible computational shortcuts in brain emulation. Whether such reductions can be done depends on whether the details on internal neural biophysics are important or not for network-relevant properties such as exact spike-timing. It may also be possible to apply reduction methods on sub-neural models, but the approach requires an understanding of the geometry of phase space of the system.
Efficient computation of branched nerve equations.
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=6698635&dopt=Abstract
http://www.neurophys.biomedicale.univ-paris5.fr/~graham/pdf/BG-JCNS-v8_3-2000.pdf
Simulators
There exist numerous simulation systems at present. Some of the more common are GENESIS (GEneral NEural SImulation System) (Wilson et al. 1989; Bower and Beeman 1998) and Neuron (Carnevale and Hines 2006).
Key issues for neural simulators are numerical stability, extendability and parallelizability.
The numerical methods used to integrate conductance based models need to both produce accurate approximation of solutions of the governing equations and run fast. This is made more problematic by the stiffness of some of the equations.
Most neural simulators have been designed to be easy to extend with new functions, often producing very complex software systems.
Neural simulators need to be able to run on parallel computers to reach high performance (see section below).
http://www.neuron.yale.edu/neuron/papers/jcns2006/parallel_nets_2006.pdf
data on performance of Neuron
http://www.neuron.yale.edu/neuron/papers/thensci/spacetime_rev1.pdf
Parallel Simulation
Networks, neurons and compartments are in general just linked to nearby entities and act simultaneously, making brain models naturally suited for parallel simulations. The main problem is finding the right granularity of the simulation (i.e. how many and which entities to out on each processing node) so that communications overhead is minimized, or finding communications methods that allow the nodes to communicate efficiently.
A PGENESIS simulation of 16 Purkinje cells with 4,500 compartments receiving input from 244,000 granule cells took 2.5 hours on a 128-processor Cray 3TE (nominally 76.8 GFLOPS) to calculate 2 seconds of simulated activity (Howell et al. 2000). This implies around 0.3 MFLOPS per neuron and a slowdown factor of 4500.
The so far (2006) largest simulation of a full Hodgkin-Huxley neuron network was performed on the IBM Watson Research Blue Gene supercomputer using the simulator SPLIT (Hammarlund and Ekeberg 1998; Djurfeldt et al. 2005). It was a model of cortical minicolumns, consisting of 22 million 6-compartment neurons with 11 billion synapses, with spatial delays corresponding to a 16 cm2 cortex surface and a simulation length of one second real-time. Most of the computational load was due to the synapses, each holding 3 state variables . The overall nominal computational capacity used was 11.5 TFLOPS, giving 0.5 MFLOPS per neuron or 1045 FLOPS per synapse. Simulating one second of neural activity took 5942 s . The simulation showed linear scaling in performance with the number of processors up to 4096 but began to show some (23%) overhead for 8192 processors (Djurfeldt et al. 2006).
An even larger simulation with 1e11 neurons and 1e15 synapses was done in 2005 by Eugene M. Izhikevich on a Beowulf cluster with 27 3 GHz processors (Izhikevich 2005). This was achieved by not storing the synaptic connectivity but by generating it whenever it was needed, making this model rather ill suited for brain emulation. One second of simulation took 50 days, giving a slowdown factor of 4.2 million.
SPLIT abstracts underlying hardware,
Djurfeldt, M., Johansson, C., ¨Orjan Ekeberg, Rehn, M., Lundqvist, M., and
Lansner, A. (2005). Massively parallel simulation of brain-scale neuronal network
models. Technical Report TRITA-NA-P0513, CSC, KTH, Stockholm.
Well implemented simulations tends to scale linearly with number of processors, although various memory and communications bottlenecks may occur and optimal use of caching can give even superlinear speedup for some problem sizes (Djurfeldt et al. 2005; Migliore et al. 2006). The main problem appears to be high connectivity, since inter-processor communications is a major bottleneck. Keeping communications to a minimum, for example by only sending information about when and where a spike has occurred, improves performance significantly. If brain emulation requires more information than this to flow between processing nodes performance will be lower than these examples.
Simulations can be time-driven and event-driven. A time-driven simulation advances one timestep at a time while an event-driven simulation keeps a queue of future events (such as synaptic spikes arriving) and advances directly to the next. For some neural network models such as integrate-and-fire, the dynamics between spikes can be calculated exactly, allowing the simulation efficiently just jump forward in time until the next spike occurs (Mattia and Giudice 2000). However, for highly connected networks the time between the arrivals of spikes become very short, and time-driven simulations are equally efficient. On the other hand, the timestep for time-driven models must be short enough that the discretization of spike timing to particular timesteps does not disrupt timing patterns, or various techniques for keeping sub-timestep timing information in the simulation (Morrison et al. 2005).
Computational Demands
A volume-based simulation where the brain is divided into size r voxels would encompass 1.4e-3/r^3 voxels. Each voxel would contain information about which cells, compartments and other information that existed inside, as well as a list of the dynamical variables (local electric fields, chemical concentrations) and local parameter values.
For 10 μm side voxels there would be 1.4e18 voxels in a human brain.
A compartment simulation of N neurons with C compartments each would have NC compartments, each storing a list of neighbor compartments, dynamical variables and local parameters. Synapses can be treated as regular compartments with extra information about weight, neurotransmittors and internal chemical state.
A fine resolution compartment model of each neuron would at least have a compartment for each synapse, making C on the order of 10^3. That would imply 1e14 compartments.
Sizes of compartments in current simulations are usually set by taking the length constants of neuron membranes into account: simulating on a much finer resolution is not needed (except possibly to deal with branching). However, for spiny cells synaptic spines likely need to be their own compartments.
PLAUSIBLE RESOLUTION
BITS FOR MODEL
FLOPS
COMMUNICATION
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