Particle diffusion is generated from Einstein's equations on Brownian motion. This allows the model to generate real-world diffusion at rates that are empirically relevant. There are currently 5 different regions in the model that can each independently scale the diffusion rate: the extrasynaptic space (ES), post-synaptic density 1 (PSD-1), post-synaptic density 2 (PSD-2) and the perisynaptic PSD-1 region (pPSD-1) and PSD-2 region (pPSD-2). The PSD and pPSD diffusion rates (D<sub>psd</sub>) can be automatically scaled in real-time by the number of PSD-95 SAP molecules currently expressed in a PSD-cluster region. For most simulations the starting SAP cluster size is 7x7 yielding 49 total SAP molecules. The amount of SAP dynamically fluctuates. It can hold a fairly steady number of about 50 SAPs, but it can also be made to grow and shrink to values ranging from 10 to 100 SAPs. The PSD diffusion rate can be scaled from these SAP values. The function for this scalar can be seen to the left. Given a range of 10 to 100 SAPs, the PSD diffusion rate values will range from 0.03 um²/s - 0.003 um²/s.
Particle diffusion is generated from Einstein's equations on Brownian motion. This allows the model to generate real-world diffusion at rates that are empirically relevant. There are currently 5 different regions in the model that can each independently scale the diffusion rate: the extrasynaptic space (ES), post-synaptic density 1 (PSD-1), post-synaptic density 2 (PSD-2) and the perisynaptic PSD-1 region (pPSD-1) and PSD-2 region (pPSD-2). The PSD and pPSD diffusion rates (D<sub>psd</sub>) can be automatically scaled in real-time by the number of PSD-95 SAP molecules currently expressed in a PSD-cluster region. For most simulations the starting SAP cluster size is 7x7 yielding 49 total SAP molecules. The amount of SAP dynamically fluctuates. It can hold a fairly steady number of about 50 SAPs, but it can also be made to grow and shrink to values ranging from 10 to 100 SAPs. The PSD diffusion rate can be scaled from these SAP values. The function for this scalar can be seen to the left. Given a range of 10 to 100 SAPs, the PSD diffusion rate values will range from 0.03 um²/s - 0.003 um²/s.
Revision as of 22:36, 3 September 2013
ReDiClus - Receptor Diffusion & Cluster Model
Simulation Space
ReDiClus Model Space
ReDiClus is simulated on a 2D surface in 3D space
The surface area represents a dendritic membrane with two synaptic spines
Baseline dimensions are scaled to real-world values
these values are based on empirical observations of distal dendrites
base dimensions are set to 60x30 units
Scale
1 unit ≃ 100 nm
10 units ≃ 1 µm
2D space: 2.3 µm x 4.6 um
PSD: 0.3 x 0.3 µm
peri-PSD: 0.3 x 0.3 µm
PSD separation: 2.0 µm
The Z axis is only 2 levels: 0 and 1
1 represents the membrane surface
0 represents intracellular space
Particle Types
There are 2 types of particles in the simulation
'Red' particle dots represent AMPA receptors
Red dots can randomly diffuse anywhere on the X-Y plane
Red dots only diffuse on the surface Z = 0
'Blue' particle dots represent PSD-95 molecules
Blue dots are contained in predefined PSD areas and cannot leave
Blue dots can exist at the surface Z = 0 or intracellularly Z = -1
Particle diffusion is generated from Einstein's equations on Brownian motion. This allows the model to generate real-world diffusion at rates that are empirically relevant. There are currently 5 different regions in the model that can each independently scale the diffusion rate: the extrasynaptic space (ES), post-synaptic density 1 (PSD-1), post-synaptic density 2 (PSD-2) and the perisynaptic PSD-1 region (pPSD-1) and PSD-2 region (pPSD-2). The PSD and pPSD diffusion rates (Dpsd) can be automatically scaled in real-time by the number of PSD-95 SAP molecules currently expressed in a PSD-cluster region. For most simulations the starting SAP cluster size is 7x7 yielding 49 total SAP molecules. The amount of SAP dynamically fluctuates. It can hold a fairly steady number of about 50 SAPs, but it can also be made to grow and shrink to values ranging from 10 to 100 SAPs. The PSD diffusion rate can be scaled from these SAP values. The function for this scalar can be seen to the left. Given a range of 10 to 100 SAPs, the PSD diffusion rate values will range from 0.03 um²/s - 0.003 um²/s.
Base ExtraSynaptic Diffusion rate D (Des)
Des ≃ 0.3 um²/s
Base PSD Diffusion rates (Dpsd)
Dpsd ≃ 0.03 um²/s
↑ to ↓
Dpsd ≃ 0.003 um²/s
Dpsd SAP scalar function
Dpsd ≃ Des/SAP
Dpsd ≃ 0.3/10 ≃ 0.03 um²/s
↑ to ↓
Dpsd ≃ 0.3/100 ≃ 0.003 um²/s
Physical Properties
ReDiClus Physics
two independent processes
In this model, there are two independently occurring processes.
1. Blue dots can be expressed at the surface or internalized within their PSD area
The Blue dot internalization/externalization rate properties are set by the Shouval cluster model equations.
2. Red dots diffuse on the X-Y plane with brownian motion
Each Red dot has an initial step size randomly drawn from a normal distribution with a mean = 1 and sd = .2
The step size for Red dots is dynamically altered when it's located in a PSD area
In a PSD, the step size is reduced by a by some factor based on the number of Blue dots currently expressed at the surface of that PSD
The more Blue dots at the surface, the more the step size is reduced
The current step size function is:
f(Rstep) = R * (10*(1 ⁄ Bn))
where Rstep is the baseline Red dot step size
where Bn is number of Blue dots currently expressed at the PSD surface
Several screen shots of the dynamic graphs in the model
FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
MEAN SQUARED DISPLACEMENT
SEE POPUP{{#info: REDICLUSANIMATION SEE ANIMATION }}
Brownian Motion Mean Squared Displacement
The goal of this calculation is to relate the simulated particle diffusion to real world values, namely velocity.
Particle velocity will be a function of MSD x units ²⁄s which scales on space (units) and time (s) parameters.
Space and time in the model are defined arbitrarily as Step_Size and Step where each Step a particle moves a distance randomly chosen from a normal distribution (µ=1,σ=.2)
a step size of 1 unit/step will produce a brownian motion MSD of ~0.52 ±0.2 units ²/s
empirical observations show that reasonable values for MSD are:
PSD 0.01 µm ²/s
synaptic 0.05 µm ²/s
extrasynaptic 0.1 µm ²/s
given an MSD of 0.52 ±0.2 units ²/s at the current parameters: 1 step = 1 unit (at µ=1,σ=.2), the model will need to be scaled such that particles move at an extrasynaptic rate of 0.1 µm ²/s.
spines are on average 1 to 10 µm apart, if the model is comparing two spines 1 µm apart, they should be separated by 5 units of model space. This is because the current particle diffusion rate of the model is .5 µm ²/s and the empirical MSD is .1 µm ²/s
Spine morphology FIG: {{#info: 3D reconstruction of a proximal CA3 pyramidal cell dendrite (blue) and a large mossy fiber bouton (translucent yellow). The cut-away in C2 shows synapses (red) onto multiple dendritic spines, some of which are highly branched. The bouton also forms nonsynaptic cell adhesion junctions (fuchsia). CLICK AWAY FROM IMAGE TO CLOSE }}
Hippocampal dendrite FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
Choquet 2007 Real Time Receptor Diffusion
{{{2}}}
Choquet 2007 Real Time Receptor Diffusion Analysis
The video represents a 10µm × 10µm section scaled to a 535px × 535px video.
1µm : 53.5px
The analysis below documents one instance of Qdot diffusion, between the 6s-7s time points.
This instance was chosen because of the clarity of motion and no Qdot flicker.
The Qdot (center) moves from pixel location (X:291, Y:302) at 6.78s to (X:319, Y346) at 6.98s
That is a distance of 52.2px in 200ms
Qdot velocity: Qv ≈ 1µm ⁄ 200ms
Note this diffusion rate of 5µm/s is 10-fold higher than the median diffusion rate reported above.
An upper bound of 5µm/s means that receptors can move between synapses in fractions of a second.
Figures:
FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}
Receptor Diffusion Rate Best Estimates
GABAA: .01 - .05 µm2/s FIG: {{#info: Choquet 2010 CLICK AWAY FROM IMAGE TO CLOSE }}
Brain Data: Facts and Figures
Estimated Number of Neurons in the Brain of Humans & Other Animals
Average number of neurons and synapses in the human brain
The brain's weight composes ~2% of total body weight (150 pound human)
Average brain width = 140 mm
Average brain length = 167 mm
Average brain height = 93 mm
Intracranial contents by volume (1,700 ml, 100%): brain = 1,400 ml (80%); blood = 150 ml (10%); cerebrospinal fluid = 150 ml (10%)
(from Rengachary, S.S. and Ellenbogen, R.G., editors, Principles of Neurosurgery, Edinburgh: Elsevier Mosby, 2005)
Average number of neurons in the brain = 100 billion
Number of neurons in octopus brain = 300 million (from How Animals See, S. Sinclair, 1985)
Number of neurons in honey bee brain = 950,000 (from Menzel, R. and Giurfa, M., Cognitive architecture of a mini-brain: the honeybee, Trd. Cog. Sci., 5:62-71, 2001.)
Number of neurons in Aplysia nervous system = 18,000-20,000
Number of neurons in each segmental ganglia in the leech = 350
Volume of the brain of a locust = 6mm3 (from The Neurobiology of the Insect Brain, Burrows, M., 1996)
Ratio of the volume of grey matter to white matter in the cerebral hemipheres (20 yrs. old) = 1.3 (Miller, A.K., Alston, R.L. and Corsellis, J.A., Variation with age in the volumes of grey and white matter in the cerebral hemispheres of man: measurements with an image analyser, Neuropathol Appl Neurobiol., 6:119-132, 1980)
Ratio of the volume of grey matter to white matter in the cerebral hemipheres (50 yrs. old) = 1.1 (Miller et al., 1980)
Ratio of the volume of grey matter to white matter in the cerebral hemipheres (100 yrs. old) = 1.5 (Miller et al., 1980)
% of cerebral oxygen consumption by white matter = 6%
% of cerebral oxygen consumption by gray matter = 94%
Average number of glial cells in brain = 10-50 times the number of neurons (New research suggests the neuron-to-glia ratio may be much smaller, closer to 1:1)
(For more information about the number of neurons in the brain, see R.W. Williams and K. Herrup, Ann. Review Neuroscience, 11:423-453, 1988)
Number of neocortical neurons (females) = 19.3 billion (Pakkenberg, B., Pelvig, D., Marner,L., Bundgaard, M.J., Gundersen, H.J.G., Nyengaard, J.R. and Regeur, L. Aging and the human neocortex. Exp. Gerontology, 38:95-99, 2003 and Pakkenberg, B. and Gundersen, H.J.G. Neocortical neuron number in humans: effect of sex and age. J. Comp. Neurology, 384:312-320, 1997.)
Number of neocortical neurons (males) = 22.8 billion (Pakkenberg et al., 1997; 2003)
Average loss of neocortical neurons = 85,000 per day (~31 million per year) (Pakkenberg et al., 1997; 2003)
Average loss of neocortical neurons = 1 per second (Pakkenberg et al., 1997; 2003)
Average number of neocortical glial cells (young adults ) = 39 billion (Pakkenberg et al., 1997; 2003)
Average number of neocortical glial cells (older adults) =36 billion (Pakkenberg et al., 1997; 2003)
Number of neurons in cerebral cortex (rat) = 21 million (Korbo, L., et al., J. Neurosci Methods, 31:93-100, 1990)
Length of myelinated nerve fibers in brain = 150,000-180,000 km (Pakkenberg et al., 1997; 2003)
Number of synapses in cortex = 0.15 quadrillion (Pakkenberg et al., 1997; 2003)
Difference number of neurons in the right and left hemispheres = 186 million MORE neurons on left side than right side (Pakkenberg et al., 1997; 2003)
Proportion by Volume (%)
Rat vs Human
Cerebral Cortex: 31 vs 77
Diencephalon: 7 vs 4
Midbrain: 6 vs 4
Hindbrain: 7 vs 2
Cerebellum: 10 vs 10
Spinal Cord: 35 vs 2
(Reference: Trends in Neurosciences, 18:471-474, 1995)
Total surface area of the cerebral cortex = 2,500 cm2 (2.5 ft2; A. Peters, and E.G. Jones, Cerebral Cortex, 1984)
Total surface area of the cerebral cortex (lesser shrew) = 0.8 cm2
Total surface area of the cerebral cortex (rat) = 6 cm2
Total surface area of the cerebral cortex (cat) = 83 cm2
Total surface area of the cerebral cortex (African elephant) = 6,300 cm2
Total surface area of the cerebral cortex (Bottlenosed dolphin) = 3,745 cm2 (S.H. Ridgway, The Cetacean Central Nervous System, p. 221)
Total surface area of the cerebral cortex (pilot whale) = 5,800 cm2
Total surface area of the cerebral cortex (false killer whale) = 7,400 cm2
(Reference for surface area figures: Nieuwenhuys, R., Ten Donkelaar, H.J. and Nicholson, C., The Central nervous System of Vertebrates, Vol. 3, Berlin: Springer, 1998)
Total number of neurons in cerebral cortex = 10 billion
(from G.M. Shepherd, The Synaptic Organization of the Brain, 1998, p. 6). However, C. Koch lists the total number of neurons in the cerebral cortex at 20 billion (Biophysics of Computation. Information Processing in Single Neurons, New York: Oxford Univ. Press, 1999, page 87).
Total number of synapses in cerebral cortex = 60 trillion (yes, trillion)
(from G.M. Shepherd, The Synaptic Organization of the Brain, 1998, p. 6). However, C. Koch lists the total synapses in the cerebral cortex at 240 trillion (Biophysics of Computation. Information Processing in Single Neurons, New York: Oxford Univ. Press, 1999, page 87).
(Caviness Jr., et al. Cerebral Cortex, 8:372-384, 1998.)
Number of cortical layers = 6
Thickness of cerebral cortex = 1.5-4.5 mm
Thickness of cerebral cortex (Bottlenosed dolphin) = 1.3-1.8 mm (S.H. Ridgway, The Cetacean Central Nervous System, p. 221)
Rate of neuron growth (early pregnancy) = 250,000 neurons/minute
Length of spiny terminals of a Purkinje cell = 40,700 micron
Number spines on a Purkinje cell dendritic branchlet = 61,000
Surface area of cerebellar cortex = 50,000 mm2 (from G.M. Shepherd, The Synaptic Organization of the Brain, 2004, p. 271)
Weight of adult cerebellum = 150 grams (Afifi, A.K. and Bergman, R.A., Functional Neuroanatomy, New York: McGraw-Hill, 1998)
Number of Purkinje cells = 15-26 million
Number of synapses made on a Purkinje cell = up to 200,000
Weight of hypothalamus = 4 g
Volume of suprachiasmatic nucleus = 0.3 mm3
Number of fibers in pyramidal tract above decussation = 1,100,000
Number of fibers in corpus callosum = 250,000,000
Area of the corpus callosum (midsagittal section) = 6.2 cm2
Species
Cerebellum Weight (grams) vs Body Weight (grams)
Human: 142 vs 60,000
Mouse: 0.09 vs 58
Bat: 0.09 vs 30
Flying Fox: 0.3 vs 130
Pigeon: 0.4 vs 500
Guinea Pig: 0.9 vs 485
Squirrel: 1.5 vs 350
Chinchilla: 1.7 vs 500
Rabbit: 1.9 vs 1,800
Hare: 2.3 vs 3,000
Cat: 5.3 vs 3,500
Dog: 6.0 vs 3,500
Macaque: 7.8 vs 6,000
Sheep: 21.5 vs 25,000
Bovine: 35.7 vs 300,000
Source: Sultan, F. and Braitenberg, V. Shapes and sizes of different mammalian cerebella. A study in quantitative comparative neuroanatomy. J. Hirnforsch., 34:79-92, 1993.
Neurons
Mass of a large sensory neuron = 10-6gram (from Groves and Rebec, Introduction to Biological Psychology, 3rd edition, Dubuque: Wm.C. Brown Publ., 1988)
Number of synapses for a "typical" neuron = 1,000 to 10,000
Diameter of neuron = 4 micron (granule cell) to 100 micron (motor neuron in cord)
Diameter of neuron nucleus = 3 to 18 micron
Length of Giraffe primary afferent axon (from toe to neck) = 15 feet
Resting potential of squid giant axon = -70 mV
Conduction velocity of action potential = 0.6-120 m/s (1.2-250 miles/hr)
Single sodium pump maximum transport rate = 200 Na ions/sec; 130 K ions/sec
Typical number of sodium pumps = 1000 pumps/micron2 of membrane surface (from Willis and Grossman, Medical Neurobiology, Mosby, St. Louis, 1981, p. 36)
Total number of sodium pumps for a small neuron = 1 million
Density of sodium channels (squid giant axon) = 300 per micron2 (from Hille, B., Ionic Channels of Excitable Membranes, Sinauer, Sunderland, 1984, p. 210.)
Number of voltage-gated sodium channels at each node = 1,000 to 2,000 per micron2 (from Nolte, J., The Human Brain, Mosby, 1999, p. 163.)
Number of voltage-gated sodium channels between nodes = 25 per micron2 (from Nolte, J., The Human Brain, Mosby, 1999, p. 163.)
Number of voltage-gated sodium channels in unmyelinated axon = 100 to 200 per micron2 (from Nolte, J., The Human Brain, Mosby, 1999, p. 163.)
Diameter of ion channel = 0.5 nanometer (Breedlove et al., Biological Psychology, 2007)
Diameter of microtubule = 20 nanometer
Diameter of microfilament = 5 nanometer
Diameter of neurofilament = 10 nanometer
Thickness of neuronal membrane = 5 nanometer (Breedlove et al., Biological Psychology, 2007)
Thickness of squid giant axon membrane = 50-100 A
Membrane surface area of a typical neuron = 250,000 um2 (Bear et al., 2001)
Membrane surface area of 100 billion neurons = 25,000 m2, the size of four soccer fields (Bear, M.F., Connors, B.W. and Pradiso, M.A., Neuroscience: Exploring the Brain, 2nd edition)
Typical synaptic cleft distance = 20-40 nanometers across (from Kandel et al., 2000, p. 176)
% neurons stained by the Golgi method = 5%
Slow axoplasmic transport rate = 0.2-4 mm/day (actin, tubulin)
Intermediate axoplasmic transport rate = 15-50 mm/day (mitochondrial protein)
Fast axoplasmic transport rate = 200-400 mm/day (peptides, glyolipids)
Number of molecules of neurotransmitter in one synaptic vesicle = 5,000 (from Kandel et al., 2000, p. 277)
% composition of myelin = 70-80% lipid; 20-30% protein
Ion Concentration (mM) in Mammalian Neurons
Intracellular vs Extracellular
Potassium: 140 vs 5
Sodium: 10 vs 150
Chloride: 10 vs 100
Calcium: 0.0001 vs 1
Because of its large number of tiny granule cells, the cerebellum contains more neurons than the rest of the brain, but takes up only 10% of the total brain volume. The number of neurons in the cerebellum is related to the number of neurons in the neocortex. There are about 3.6 times as many neurons in the cerebellum as in the neocortex, a ratio that is conserved across many different mammalian species. What gives? Two things really: (1) The neocortex has a large proportion of pyramidal neurons which are much bigger than granule cells, and (2) Purkinje cells can make up to 200k synaptic connections (contrast that with 1k - 10k for "typical" neurons of other types).
A microprocessor incorporates the functions of a computer's central processing unit on a single integrated circuit. It is a multipurpose, programmable device that accepts digital data as input, processes it according to instructions stored in its memory, and provides results as output.
