Abstract
Identifying the regime in which the cortical microcircuit operates is a prerequisite to determine the mechanisms that mediate its response to stimulus. Classic modeling work has started to characterize this regime through the study of perturbations, but an encompassing perspective that links the full ensemble of the network’s response to appropriate descriptors of the cortical operating regime is still lacking. Here we develop a class of mathematically tractable models that exactly describe the modulation of the distribution of cell-type-specific calcium-imaging activity with the contrast of a visual stimulus. The model’s fit recovers signatures of the connectivity structure found in mouse visual cortex. Analysis of this structure subsequently reveal parameter-independent relations between the responses of different cell types to perturbations and each interneuron’s role in circuit-stabilization. Leveraging recent theoretical approaches, we derive explicit expressions for the distribution of responses to partial perturbations which reveal a novel, counter-intuitive effect in the sign of response functions.
Competing Interest Statement
The authors have declared no competing interest.
Nomenclature
- αi
- Short for population to which cell i belongs
- χ
- Linear response matrix of the low-dimensional circuit
- Δα
- Variance of the input to population α
- κ and ν
- low rank vectors that compose σ
- Variance in the population α
- ω
- Low-dimensional connectivity matrix
- ΠL
- Diagonal matrix with entries κ
- ΠR
- Diagonal matrix with entries v
- Σ
- Optogenetic targeting matrix
- σαβ
- matrix of the standard deviations of the weight matrix W
- τ
- Time constant
- ξ
- Power in a threshold power law input-output function
- A
- Diagonal matrix with factors to transform calcium to rates
- B
- Measuring matrix
- c
- Contrast value, usually normalized to 1
- E
- Error function
- F
- Diagonal matrix with the derivatives of f at the fixed point of the high-dimensional circuit
- f
- Input-output function /nonlinearity
- f′
- Derivative of f
- h
- External inputs to the network
- J
- Jacobian
- k
- Normalized entries of the low-dimensional linear response matrix χ
- mα
- Mean firing rate in population α for high-D model
- N
- Number of neurons in the high-D system
- n
- Number of populations (different cell-types) in the network
- Nα
- Number of neurons in population α
- Pα
- Distribution of activity over population α
- qα
- Fraction of cells in population α : Nα /N
- R
- Linear response of the high-D system
- r
- Activity, rα is the activity in population α
- R0
- Linear response of the high-D system in the absence of disorder
- T
- Diagonal matrix of time constants
- uα
- Mean input to population α
- vα
- Second moment of the activity distributions in population α
- W
- Weight matrix of the high-dimensional model
- wαβ
- Mean connection strength form population β to population α
- Weight connecting neuron j in population β to neuron i in population α
- W0
- matrix of entries wαβ
- z
- Input current
- f′
- Diagonal matrix with the derivatives of f at the fixed point of the low-dimensional circuit
- HFP
- Homogeneous fixed point
- high-D
- High-dimensional (i.e. N dimensional) model, with 4 populations
- low-D
- Low-dimensional (i.e. 4-dimensional) model
