Fig 1.
General characteristics of models, and the mappings between them, including steps for network parameters (solid arrows) and cellular parameters (dashed arrows).
Arrows that arise from, or terminate at, multiple source or destination models, respectively, indicate a common characteristic of the models specific for a given mapping. For explanation of model abbreviations, see Table 1.
Table 1.
Abbreviations: 2D, 2 dimensional; HH, Hodgkin-Huxley; LIF, leaky integrate-and-fire; FR, firing rate; Multi-Compt E, multiple compartment excitatory neuron.
Fig 2.
Cortical domain containing 4 hypercolumns that provides the reference geometry for the 2D model A; the dots mark the pinwheel centers; the colors mark the orientations coded by bars in the legend.
To obtain an activity profile in orientation space the 2D model activity is projected to the ring shown at the lower left quadrant.
Fig 3.
A. Reduction of 2D pinwheel geometry to the ring-geometry requires integration over the radial vectors with the angle ξ−θ between them. B. Mapping 2D pinwheel geometry to the ring, showing the weight function w0(ϕ) according to the Eqs 21 and 22 (solid line), and its approximations of the forms 1 + kcos(ϕ) (dotted line) and b + a exp(−|ϕ|/ϕr) (dashed line) for the case of R = d/2.
Fig 4.
Comparative spike train responses of the mapped single neuron models: HH to LIF, and 2 to 1 compartment LIF.
A) The spike train for the non-adaptive 2-compartmental HH pyramidal neuron (solid line), the 2 compartment LIF model (dashed line) and the 1 compartment LIF model (dotted line), the latter two with . The input to the dendrite of the two compartment models was I = 1.2nA and S = 0.035 μS; according to the ratio of the dendritic and somatic input conductances, Gin,d/Gin, the current and conductance input to the 1-compartmental neuron was adjusted to 0.62nA and 0.018μS, respectively. B) The spike train for the HH interneuron (solid line) and the corresponding LIF model with
(dashed line), in response to I = 0.2nA and S = 0.
Fig 5.
Comparisons of threshold-linear approximations for noisy LIF and HH excitatory cell models.
Threshold-linear approximation (dashed line) of the LIF noisy neuron firing rate (Eq 5, solid line). To compare, the response of a LIF neuron with no noise is shown by the dotted line. The steady-state rate of the adaptive neuron described in Section 4.3 is shown by red dots, each dot corresponding to one stimulation current amplitude and being obtained by Monte-Carlo simulation with explicit noise during 2 seconds. The dashed-dotted line is the threshold-linear approximation with a gain of 0.026 Hz/pA.
Fig 6.
The response of model A (HH 2E EI Exp 2D) over the cortical surface (1mm2 containing 4 hypercolumns; the dots mark the pinwheel centers) to a 0° oriented stimulus presented at t = 0, and shifting to the 45° orientation at t = 100ms.
The circle at the lower left quadrant of the input maps correspond to the activity profile of this model shown in the first panel of Fig 8.
Fig 7.
Evolution of voltage and firing rate profiles of model A (HH 2E EI Exp 2D) as a function of preferred orientation, in response to the stimulus sequence described in Fig 6.
A, sub-threshold voltage for the excitatory population; B, sub-threshold voltage for the inhibitory population; C, inhibitory population firing rate; D, excitatory population membrane potential across horizontal line across the cortical surface passing through two pinwheel centers at two time moments t = 90 and 190ms, corresponding to the snapshots in Fig 6; E, excitatory population firing rate across the same line at the same time moments.
Fig 8.
Responses of a hypercolumn as a function of preferred orientation to the stimulus sequence described in Fig 6, for the excitatory population of the various models listed in Table 1 (ref. “Thalamic” input panel). The activity for model A corresponds to the circle indicated in Fig 6.
Fig 9.
The tuning curve for the input compared to steady-state solutions for the firing rate calculated for each of the models of Table 1, obtained as cross-sections at t = 300ms of the plots shown in Fig 8 (un-scaled in A, and normalized in B).
The profile with the smallest amplitude corresponds to the simulation by the “adaptive” firing-rate ring model G as described in the text and indicated in Fig 10. The half-widths at half-maximums are 38° for model A, 36° for model B, 31° for model C, 28° for model D, 27° for models E and F, 24° for model G, and 22° for model G with the gain rescaled to account for adaptation.
Fig 10.
The diagram of the steady-state solutions of the canonical firing-rate ring model G on the plane of its parameters J0 and J1 (adapted from [2]).
Amplitude instability corresponds to the state where the activity in the ring increases without any possibility to regulate it. Homogeneous phase (“feedforward” hypothesis) corresponds to a state of weak interactions. The activity in the ring follows directly from the thalamic input, apart from a threshold non-linearity. Marginal phase (“recurrent” hypothesis) corresponds to a state where only a tuned activity profile is stable, partially but not completely determined by the input shape and dynamics. This state occurs for sufficiently strong recurrent tuned inputs (J1) and, to a lesser extent, with sufficiently strong inhibition (J0). The small square marks the parameters found by the mapping expressions for model G (I0 = −26.5, I1 = 44, J0 = −0.52, J1 = 2.8). The small circle in the homogeneous state corresponds to a variation of model G where the value of was derived from the adaptive HH excitatory cell model (I0 = −3.55, I1 = 7.4, J0 = −0.063, J1 = 0.46).
Fig 11.
Contrast invariance for the excitatory population firing rate (red) in models D (left) and A (right), compared with the responses (green) of feed-forward versions of each model () to distinguish the contribution of intracortical pathways.
The “weak” and “high” contrast values were 0.4 and 0.8, respectively. As described in the text, the higher gain of model D as compared to model A, as seen in the different response magnitudes, account for the marginal phase behavior of the former, thus contrast-invariance, and the homogeneous phase behavior of the latter, thus no contrast-invariance. The intracortical connections serve to increase input-output sharpening for both models.
Fig 12.
The traces of the center of mass of the population firing rate over time, calculated for all models, according to Figs 7 and 8.
Note the tilt after-effect, thus an overshoot of the response tuning, and virtual rotation, thus a lag in the tuning, after the abrupt shift in stimulus orientation at 100ms.
Fig 13.
The firing of a population of noisy LIF neurons in response to current step stimulation.
From top to bottom: Current step. Spike raster plot for for 4 neurons. Voltage traces for the 4 neurons. Population firing rate calculated by five models: Monte-Carlo simulation of 4000 individual trials of a noisy LIF neuron, the firing-rate (FR) model based on Eq 2, the firing-rate (FR) model with τFR based on Eq 1, an infinite population of noisy LIF neurons evaluated with the KFP approach (Eqs 53 and 54) and the conductance-based refractory density (CBRD) approach (Eqs 55 and 57). The response of the FR with shunt cell model (not shown) is identical to the current-based FR cell model, since in both cases the transient response is fixed by τFR, and the threshold-linear transfer function of the latter was fit to the steady-state firing rate of the LIF model, that in turn defines the former.
Fig 14.
The steady-state firing rate of a population of LIF neurons as a function of the strength of the input current stimulation, with synaptic conductance S set to 0 and to 2gL, with and without noise (C = 0.1nF, gL = 10nS, VTh − VRest = 10mV, VReset = VRest, τm = 10ms, σI = 28pA), according to Eq 5, compared with the result from the CBRD evaluation with noise.