Exploring the parameter spaces of neural models
These data were used in
The unexplored territory of neural models: Potential guides for exploring the function of metabotropic neuromodulation by Michael Hasselmo, Andrew Alexander, Alec Hoyland, Jennifer C Robinson, Ausra Saudargiene, Holger Dannenberg (2020, in review at Neuroscience).
See a PDF writeup of the meta-analysis here.
Reticular thalamus (RE) cells were connected with GABAergic synapses of the A or B subtypes. Neurons were arrayed on a 2-dimensional lattice.
The model is conductance-based, with 6 intrinsic currents, synaptic input, and a calcium buffering mechanism. This results in 9 state variables per cell, plus 1 state variable per synapse. There are 16 parameters per cell.
The following experiments were performed:
- 2 RE cells, reciprocal GABA_A inhibition (2 cells, 2 synapses)
- 2 RE cells, reciprocal GABA_A inhibition and self-connection (2 cells, 4 synapses)
- 5 RE cells, nearest-neighbor reciprocal GABA_A inhibition (5 cells, 10 synapses)
- 5 RE cells, nearest-neighbor reciprocal GABA_A inhibition and self-connections (5 cells, 15 synapses)
- 5 RE cells, fully connected with GABA_A synapses (5 cells, 20 synapses)
- 4 RE cells, fully-connected with GABA_A synapses and self-connections (5 cells, 25 synapses)
- 100 RE cells, nearest-neighbor GABA_A synapses (100 cells, 400 synapses)
- 100 RE cells, dense proximal GABA_A synapses including autapses (100 cells, 2,500 synapses)
- 400 RE cells, dense proximal GABA_A synapses including autapses (400 cells, 10,000 synapses)
- 1,600 RE cells, dense proximal GABA_A synapses, including autapses (1,600 cells, 40,000 synapses)
All experiments were repeated with GABA_B synapses, though only a subset of experiments were shown.
This paper investigated subthreshold synaptic summation in a model CA1 pyramidal cell. The model is "n123" from the Duke/Southampton archive of neuronal morphology, with an axon added. It contains 183 compartments. The number of synapses was varied from 2 to 28.
2,756 intrinsic parameters plus 32 synaptic parameters.
The neurons here are a reduced Hodgkin-Huxley model with two state variables and auxiliary after-spike resetting. There are four parameters per cell.
Izhikevich simulates a sparse network of 10,000 cells with 1,000,000 synaptic connections. Synapses are instantaneous, with no state variables, but each synapse has a synaptic weight parameter.
Linear model with 2 state variables with 9 parameters. Since the model is a linear dynamical system, it is solvable and does not need to be numerically integrated.
Each cell is a single compartment. Thalamocortical (TC) cells have 6 intrinsic currents. Reticular thalamic (RE) cells have 4 intrinsic currents. Each cell have 2 synaptic currents: thalamocortical cells have thalamocortical GABA_A and and reticular GABA_B; reticular thalamic cells have AMPAergic current from thalamocortical inputs, and GABA_A from reticular inputs.
TC cells have 8 state variables each. RE cells have 6 state variables each. Each AMPA and GABA_A synapse contributes 1 state variable. Each GABA_B synapse contributes 2: one for fractional receptor activation and another for concentration of activated G protein.
TC cells have 16 parameters each. RC cells have 12 parameters each.
In practice, many of these parameters are degenerate. For example, the potassium reversal potential or the specific membrane capacitance is unlikely to vary too much among your cells of the same type. Therefore, the lower-bound estimate for the number of parameters is closer to one per current, or 6 parameters for TC cells and 4 parameters for RE cells.
The following networks were tested:
- 1 TC cell
- 2 TC cells and 2 RE cells with 6 GABA_A synapses, 2 GABA_B synapses, and 4 AMPA synapses
- 2 TC cells and 2 RE cells with 2 GABA_B synapses and 4 AMPA synapses
- 50 TC cells and 50 RE cells with 1,100 GABA_A synapses, 550 GABA_B synapses, and 550 AMPA synapses
CA1 pyramidal model (183 compartments) with 17 types of voltage-dependent ion channels.
This is really quite cool. I need to look into this further.
9 parameters in the spiking neuron model (exponential integrate and fire), 8 spiking network parameters, 1600 excitatory cells, 400 inhibitory cells. 1,280 synapses on average with the same number of parameters (synaptic weight).
The model includes 14 neuron types with 74, 74, 59, 59, 59, 59, 61, 61, 50, 59, 59, 59, 137, and 59 compartments. There are 3,560 neurons total and 229,940 compartments.
There are 11 conductances in each compartment. The neuron reconstruction has a specific membrane capacitance, diameter, and length for each compartment. Each compartment must also have a calcium buffering mechanism. Therefore, the number of intrinsic parameters is 3,679,051. This is a 6 layer model.
There are 2 synapse types, for 2 more synaptic parameters, plus synaptic weights. There are 148,020 synapses from superficial RS pyramidal cells. There are 11,755 synapses from superficial FRB pyramidal cells. There are 31,200 synapses from superficial basket cells. There are 23,550 synapses from superficial axoaxonic cells. There are 69,300 synapses from superficial LTS interneurons. There are 74,100 synapses from spiny stellate cells. There are 53,100 synapses from tufted IB pyramidal cells. ... There are about 500,000 synapses.
4 layer model with 80,000 cells, 300,000,000 synapses. Leaky integrate-and-fire model with 5 parameters each. The synapse model is current-based, with a decay constant and a synaptic weight. Therefore, there are 400,000 intrinsic parameters and 300,000,001 synaptic parameters.
3 neurons, 9 compartments, 38 parameters, 7 synaptic parameters.
2 neurons, 20 intrinsic parameters, 8 synaptic parameters.
8 neurons, 88 intrinsic parameters, 33 synaptic parameters.
200 neurons, single-compartment, with Hodgkin-Huxley conductances. 1003 intrinsic parameters, 21,600 synapses.
100 neurons, single-compartment, with Hodgkin-Huxley conductances. 503 intrinsic parameters, 9,900 synapses.