igif model  (Nonlinear Dynamics)

Journal: PLoS Computational Biology

Article Title: Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons

doi: 10.1371/journal.pcbi.1004761

Figure Lengend Snippet: (A) Schematic representation of the iGIF model. The input current is first low-pass filtered by the Passive membrane filter κ m ( t ) = Θ ( t ) C - 1 e - t τ m . The resulting signal models the subthreshold membrane potential V ( t ) and, after subtraction of the firing threshold V T ( t ), is transformed into a firing intensity λ( t ) by the exponential Escape-rate nonlinearity . Spikes are emitted stochastically and elicit both a Spike-triggered conductance η ( t ) and a Spike-triggered threshold movement γ ( t ). In the iGIF model, but not in the GIF model, the firing threshold V T ( t ) is coupled to the subthreshold membrane potential (dashed circuit). For that, the membrane potential V ( t ) is first passed through the nonlinear Threshold coupling function θ ∞ ( V ) and then low-pass filtered by the Threshold filter κ θ ( t ) = Θ ( t ) τ θ - 1 e - t τ θ . (B)-(E) Average parameters extracted from 6 Pyr neurons. Black: iGIF-free, red: iGIF-Na. Gray areas indicate one standard deviation across cells for the iGIF-Na model. (B) Passive membrane filter κ m ( t ). Inset: passive membrane timescale τ m . Open circles: results from individual cells. Bar plot: mean and standard deviation. (C) Spike-triggered conductance η ( t ). Inset: same data on log-log scales. (D) Spike-triggered threshold movement γ ( t ). (E) Nonlinear threshold coupling θ ∞ ( V ) (iGIF-free, solid black line; iGIF-Na, solid red line). Spikes are emitted stochastically when the spiking boundary V = V T is approached. This boundary defines the line where the probability p of emitting a spike during a time bin of Δ T = 0.1 ms reaches p = 0.63. To the right, the probability increases further. In the absence of previous action potentials, the spiking boundary is given by V = θ (dashed black). After an action potential, the spiking boundary instantaneously shifts to the right by γ (0)≈25 mV (dashed gray), and then slowly decays back to V = θ . After each spike, the variables ( V ( t ), θ ( t )) are reset to ( V reset , V T * ) (open circle; error bars denote one standard deviation across cells). The open red circle indicates the Na + half-inactivation voltage V i , where the threshold becomes sensitive to the membrane potential. Inset: percentage increase in model log-likelihood ( LL ) of iGIF models with respect to the GIF model. Open circles: model performance on the same cell for iGIF-free and iGIF-Na. Bar plots: mean and standard deviation across neurons. (F) Top: LL percentage increase of the iGIF-free model as a function of τ θ . Red circle: optimal timescale τ θ . Data are shown from a typical neuron. Bottom: optimal timescales τ θ extracted from 6 Pyr neurons. (G) Top: LL percentage increase of the iGIF-Na model as a function of k i and V i . The LL increases from dark to light red. Red circle: optimal parameters. Bottom: optimal parameters extracted from 6 different Pyr neurons. Note that the half-inactivation voltages V i reported in the figure are positively biased due to uncompensated liquid junction potential of 14.5 mV (see ). The mean and the standard deviational ellipse across cells are shown in red.

Article Snippet: Analytically reducing the iGIF model to a GLM, finally showed that the nonlinear dynamics of the firing threshold adaptively shorten the effective timescale over which L5 Pyr neurons integrate their inputs, thus enhancing sensitivity to rapid input fluctuations over a broad range of input statistics (Figs and ).

Techniques: Membrane, Transformation Assay, Standard Deviation

Journal: PLoS Computational Biology

Article Title: Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons

doi: 10.1371/journal.pcbi.1004761

Figure Lengend Snippet: (A) Comparison between steady-state f − μ I curves observed in a typical Pyr neuron (gray lines) and produced by the iGIF model (colored lines). Different colors and gray levels indicate the magnitude of input fluctuations σ I (see legends). (B) Average firing threshold observed in a typical cell and produced by the iGIF model. Conventions as in panel A . Since our threshold analysis is based on relative changes rather than on absolute values , model predictions were normalized with a constant offset to have the same mean as the experimental data. (C) Summary data of results obtained in six Pyr neurons (gray) and iGIF models (red). Top: percentage change in steady-state firing rate obtained in response to the strongest DC input by increasing the input standard deviation σ I (data are presented as in ). Bottom: average change in voltage threshold obtained by increasing the input standard deviation σ I (data are presented as in ). (D)–(F) As a control, the GIF model response (yellow) is compared against data (gray). Results are presented as in panels A – C . (G) The normalized, cross-validated log-likelihood LL (see ) of the iGIF (red) and the GIF model (yellow), is shown as a function of μ I . Solid lines and gray areas indicate the average and one standard deviation across neurons, respectively. For comparison, the performance of the GLM with input-specific filters is shown in black (data reported from ). (H) Normalized, cross-validated log-likelihood LL of the iGIF model as a function of that of the GIF model. Each data point indicates the LL obtained from an individual neuron responding to a specific input ( μ I ; σ I )

Article Snippet: Analytically reducing the iGIF model to a GLM, finally showed that the nonlinear dynamics of the firing threshold adaptively shorten the effective timescale over which L5 Pyr neurons integrate their inputs, thus enhancing sensitivity to rapid input fluctuations over a broad range of input statistics (Figs and ).

Techniques: Comparison, Produced, Standard Deviation, Control

Journal: PLoS Computational Biology

Article Title: Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons

doi: 10.1371/journal.pcbi.1004761

Figure Lengend Snippet: (A) Segment of the 20-second current used in the test dataset . (B) Spiking response of a Pyr neuron (black) to 9 repetitive injections of the current shown in panel A . The predictions of the iGIF and GIF model are shown in red and yellow, respectively. (C) PSTHs computed by filtering the spike trains shown in panel B with a 500 ms rectangular window. The dashed line indicates 0 Hz. (D) Typical intracellular response (black), as well as typical iGIF (red) and GIF (yellow) model prediction, to a single presentation of a 1-s segment of the current shown in panel A . Black triangles and colored dots indicate spikes. (E) – (G) Summary data showing the performance of the iGIF (red) and the GIF (yellow) model in predicting the test dataset . Filled bars and open bars show the performance of models trained on the f – I dataset and the training dataset , respectively. Error bars represent one standard deviation across neurons. (E) Spike-timing prediction as quantified by the similarity measure M d * . The iGIF model significantly outperforms the GIF model with parameters extracted from the f – I dataset ( M d * = 0 . 75 , s.d. 0.03, iGIF; M d * = 0 . 49 , s.d. 0.08, GIF; n = 6, paired Student t -test, t 5 = −8.44, p = 3.8 · 10 −4 ) and from the training dataset ( M d * = 0 . 83 , s.d. 0.02, iGIF; M d * = 0 . 76 , s.d. 0.05, GIF; n = 6, paired Student t -test, t 5 = −3.25, p = 0.022). (F) The prediction error ϵ PSTH on the PSTH (see panel C ) was quantified by computing the root mean square error between data and model prediction. The iGIF model significantly outperforms the GIF model with parameters extracted from the f – I dataset ( ϵ PSTH = 1.93, s.d. 0.71 Hz, iGIF; ϵ PSTH = 3.22, s.d. 0.91 Hz, GIF; n = 6, paired Student t -test, t 5 = 6.33, p = 1.5 · 10 −3 ) but not when the parameters are extracted from the training dataset ( ϵ PSTH = 1.41, s.d. 0.28 Hz, iGIF; ϵ PSTH = 1.52, s.d. 0.30 Hz, GIF; n = 6, paired Student t -test, t 5 = 0.95, p = 0.38). (G) Comparison between GIF the iGIF model stochasticity. The iGIF model is significantly less stochastic than the GIF model (Δ V = 0.59, s.d. 0.12 mV, iGIF; Δ V = 2.74, s.d. 0.50 mV, GIF; n = 6, paired Student t -test, t 5 = 9.68, p = 2.0 · 10 −4 , f – I dataset ; Δ V = 0.60, s.d. 0.05 mV, iGIF; Δ V = 1.28, s.d. 0.29 mV, GIF; n = 6, paired Student t -test, t 5 = 5.71, p = 2.3 · 10 −3 , training dataset ).

Article Snippet: Analytically reducing the iGIF model to a GLM, finally showed that the nonlinear dynamics of the firing threshold adaptively shorten the effective timescale over which L5 Pyr neurons integrate their inputs, thus enhancing sensitivity to rapid input fluctuations over a broad range of input statistics (Figs and ).

Techniques: Standard Deviation, Comparison

Journal: PLoS Computational Biology

Article Title: Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons

doi: 10.1371/journal.pcbi.1004761

Figure Lengend Snippet: Dynamics of the iGIF model (with parameters extracted from a typical Pyr neuron) responding to three fluctuating currents with standard deviation σ I = 100 pA, temporal correlation τ I = 3 ms and μ I = 90 pA (panels A and D ), μ I = 230 pA (panels B and E ) and μ I = 450 pA (panels C and F ). (A) Top: input current I ( t ) (gray), membrane potential V ( t ) (black), voltage-dependent threshold adaptation θ ( t ) (red) and spike-dependent threshold adaptation ϕ ( t ) = ∑ t ^ γ ( t - t ^ ) (blue) as a function of time. The four dotted lines indicate: I = 0 nA, V = E L , θ = V T * and φ = 0 mV. Bottom: distribution of subthreshold membrane potential fluctuations P ( V ) (gray) and of voltages at which spikes were initiated P ( V |spike) (black). The instantaneous threshold-coupling gain G θ ( V ) = d d V θ ∞ Na ( V ) (red) defines the sensitivity of the firing threshold to the membrane potential and can be interpreted as an activation function for the threshold-voltage coupling. Since θ ∞ Na ( V ) is a nonlinear function of V , G θ depends on the membrane potential. For low inputs, the histogram of voltage s P ( V ) covers mostly the region V ≤ V i (dashed horizontal line), where the threshold-voltage coupling is not active. (B) Same results as in panel A obtained by increasing the mean input μ I . With medium input strength, the majority of the voltage distribution lies in the region V ≈ V i , where the threshold-voltage coupling is active. (C) As in panel B , but for strong mean input μ I . In this case, the voltage-threshold coupling is almost always active (i.e., V ≥ V i ). (D) The GLM integration filter κ GLM ( t ) (solid gray) obtained by fitting a GLM to the spiking activity of the iGIF model (panel A ) is compared against the passive membrane filter κ m ( t ) (dashed black, see ) and the effective membrane filter κ m eff ( t ) (solid black), which accounts for the conductance increase mediated by η . The GLM integration filter κ GLM ( t ) (solid gray) matches the theoretical filter κ ^ GLM ( t ) (solid red; see ), which takes into account the effect of the threshold-voltage coupling. (E)-(F) Same plots as in panel D , but for increased mean input μ I .

Article Snippet: Analytically reducing the iGIF model to a GLM, finally showed that the nonlinear dynamics of the firing threshold adaptively shorten the effective timescale over which L5 Pyr neurons integrate their inputs, thus enhancing sensitivity to rapid input fluctuations over a broad range of input statistics (Figs and ).

Techniques: Standard Deviation, Membrane, Activation Assay, Activity Assay

Journal: PLoS Computational Biology

Article Title: Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons

doi: 10.1371/journal.pcbi.1004761

Figure Lengend Snippet: (A) Average effective membrane filters κ m eff ( t ) computed with iGIF model parameters extracted from 6 Pyr neurons by increasing μ I from 0.05 nA (blue) to 0.5 nA (red, see colorbars in panel D ). The passive membrane filter κ m ( t ) (dashed black) is shown for comparison. Inset: average effective conductance g L eff as a function of μ I . The gray area indicates one standard deviation across neurons and the dashed black line indicates the passive leak conductance g L . (B) Same results as in panel A , but for the average integration filter κ ^ GLM ( t ) . Inset: average coupling strength G ¯ θ as a function of the mean input μ I . Conventions are as in panel A . (C) The effective membrane timescale τ ^ m eff (red) and the effective timescale of integration τ ^ GLM (blue) predicted by the iGIF model with parameters extracted from six neurons match the experimental data (black; copied from ). Colored lines and gray areas indicate the mean and one standard deviation across neurons. The effective timescales of integration τ ^ GLM (red) predicted by the iGIF model were obtained by fitting a single-exponential function to κ ^ GLM ( t ) . (D) The iGIF model explains the adaptive changes in the spike-history filter h GLM ( t ) (see ). Left: average spike-history filter h GLM ( t ) obtained by fitting a GLM to artificial data generated by simulating the iGIF model response to fluctuating currents of increasing μ I (see colorbar). Right: average theoretical filters h ^ GLM ( t ) computed using iGIF model parameters extracted from 6 Pyr neurons. Because of the approximations involved in the analytical derivation of h ^ GLM ( t ) , the strength of the GLM spike-history filters are underestimated during the firsts τ m eff ms (see ). (E)-(F) Switching experiment performed in a iGIF model (with parameters extracted from a typical cell) to study the temporal evolution of single neuron adaptation induced by a sudden change in μ I . (E) Top: fluctuating current (gray) generated by periodically switching μ I (dark gray) between 0.1 nA and 0.27 nA, with cycle period T cycle = 10 s (only one cycle is shown). Middle: effective timescale of integration τ ^ GLM as a function of time. Bottom: output firing rate. While spike-frequency adaptation occurs on both fast and slow timescales, changes in τ ^ GLM triggered by a switch in μ I are almost instantaneous. Horizontal black lines indicate (from top to bottom): 0 nA, 0 ms and 0 Hz. (F) Comparison between effective integration filters κ ^ GLM ( t ) estimated at different moments in time during the switching experiment (see arrows in panel E ). The filters estimated at steady-state (late low, late high; defined as the last 150 ms before the stimulus switch) closely resemble the ones estimated right after the stimulus switch (early low, early high; first 150 ms after the stimulus switch), indicating that adaptive changes in κ ^ GLM ( t ) are almost instantaneous. The passive membrane filter κ m ( t ) (dashed black) is shown for comparison. In all panels, input currents were generated according to with σ I = 100 pA and τ I = 3 ms.

Article Snippet: Analytically reducing the iGIF model to a GLM, finally showed that the nonlinear dynamics of the firing threshold adaptively shorten the effective timescale over which L5 Pyr neurons integrate their inputs, thus enhancing sensitivity to rapid input fluctuations over a broad range of input statistics (Figs and ).

Techniques: Membrane, Comparison, Standard Deviation, Generated