Journal: Frontiers in Aging Neuroscience

Article Title: Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis

doi: 10.3389/fnagi.2022.765966

Figure Lengend Snippet: Inverse mutation-induced rod degeneration rate and TF threshold concentration—Uniform target cone degeneration profile. (A) inverse mutation-induced rod degeneration rate, ϕ r inv (θ) ( f crit = 3 × 1 0 - 5 ); (B) inverse TF threshold concentration, f crit inv (θ) ( ϕ r = 7.33 × 1 0 - 2 ). The solid black and dashed green curves correspond to Scaling 1 (α = 7.01 × 10 4 and β = 1.79 × 10 6 ), while the solid blue and dashed red curves correspond to Scaling 2 (α = 7.01 × 10 2 and β = 1.79 × 10 4 ). The black and blue solid curves are analytical approximations to the inverses, obtained by plotting Equations (7) and (10), respectively (A) , and Equations (8) and (11) respectively (B) . The green and red dashed curves are numerical inverses, obtained by using the Matlab routines fminsearch (A) and patternsearch (B) to calculate the ϕ r and f crit profiles for which the contour described by p c ( θ , t ) / p ~ c ( θ ) = 0.99 matches the target cone degeneration profile, t degen (θ). Equations (1–5) were solved at each iteration using the method of lines, with 101 mesh points. Insets show magnified portions of each graph. Numerical inverses are calculated and plotted only at those locations (eccentricities) where the analytical inverse fails to generate a t degen (θ) profile matching the target profile. Inverses are monotone increasing for Scaling 1, and increase initially for Scaling 2 before reaching a maximum and decreasing toward the ora serrata (θ = 1). Numerical solutions reveal lower values of the inverses near the fovea (θ = 0) than the analytical approximations suggest. Cone degeneration profile formulas and parameters are given in . Remaining parameter values as in .

Article Snippet: To find φ r inv (θ), we use the Matlab routine fminsearch (which uses a simplex search method), while to obtain f crit inv (θ) the Matlab routine patternsearch (which uses an adaptive mesh technique) was found to be more effective.

Techniques: Mutagenesis, Concentration Assay

Journal: Frontiers in Aging Neuroscience

Article Title: Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis

doi: 10.3389/fnagi.2022.765966

Figure Lengend Snippet: Inverse mutation-induced rod degeneration rate and TF threshold concentration—Pattern 1A target cone degeneration profiles. (A,C,E) inverse mutation-induced rod degeneration rate, ϕ r inv (θ) ( f crit = 3 × 1 0 - 5 ); (B,D,F) inverse TF threshold concentration, f crit inv (θ) ( ϕ r = 7.33 × 1 0 - 2 ). (A,B) linear target cone degeneration profile, t degen (θ); (C,D) concave up quadratic t degen (θ) profile; (E,F) concave down quadratic t degen (θ) profile. The solid black and dashed green curves correspond to Scaling 1 (α = 7.01 × 10 4 and β = 1.79 × 10 6 ), while the solid blue and dashed red curves correspond to Scaling 2 (α = 7.01 × 10 2 and β = 1.79 × 10 4 ). The black and blue solid curves are analytical approximations to the inverses, obtained by plotting Equations (7) and (10), respectively, (A,C,E) and Equations (8) and (11), respectively, (B,D,F) . The green and red dashed curves are numerical inverses, obtained by using the Matlab routines fminsearch (A,C,E) , and patternsearch (B,D,F) to calculate the ϕ r and f crit profiles for which the contour described by p c ( θ , t ) / p ~ c ( θ ) = 0.99 matches the target cone degeneration profile, t degen (θ). Equations (1–5) were solved at each iteration using the method of lines, with 26, 51, or 101 mesh points. Insets show magnified portions of each graph. Numerical inverses are calculated and plotted only at those locations (eccentricities) where the analytical inverse fails to generate a t degen (θ) profile matching the target profile. Inverses are monotone increasing functions for both scalings in (A, B, E, F) , and for Scaling 1 in (C,D) while the inverses increase initially for Scaling 2 before reaching a maximum and decreasing toward the ora serrata (θ = 1) in (C,D) . Numerical solutions reveal lower values of the inverses near the fovea (θ = 0) than the analytical approximations suggest. Cone degeneration profile formulas and parameters are given in . Remaining parameter values as in .

Article Snippet: To find φ r inv (θ), we use the Matlab routine fminsearch (which uses a simplex search method), while to obtain f crit inv (θ) the Matlab routine patternsearch (which uses an adaptive mesh technique) was found to be more effective.

Techniques: Mutagenesis, Concentration Assay

Journal: Frontiers in Aging Neuroscience

Article Title: Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis

doi: 10.3389/fnagi.2022.765966

Figure Lengend Snippet: Inverse mutation-induced rod degeneration rate and TF threshold concentration—Pattern 1B target cone degeneration profiles. (A,C,E) inverse mutation-induced rod degeneration rate, ϕ r inv (θ) ( f crit = 3 × 1 0 - 5 ); (B,D,F) inverse TF threshold concentration, f crit inv (θ) ( ϕ r = 7.33 × 1 0 - 2 ). (A,B) linear target cone degeneration profile, t degen (θ); (C,D) quadratic t degen (θ) profile; (E,F) exponential t degen (θ) profile. The solid black and dashed green curves correspond to Scaling 1 (α = 7.01 × 10 4 and β = 1.79 × 10 6 ), while the solid blue and dashed red curves correspond to Scaling 2 (α = 7.01 × 10 2 and β = 1.79 × 10 4 ). The black and blue solid curves are analytical approximations to the inverses, obtained by plotting Equations (7) and (10), respectively, (A,C,E) , and Equations (8) and (11), respectively, (B,D,F) . The green and red dashed curves are numerical inverses, obtained by using the Matlab routines fminsearch (A,C,E) , and patternsearch (B,D,F) to calculate the ϕ r and f crit profiles for which the contour described by p c ( θ , t ) / p ~ c ( θ ) = 0.99 matches the target cone degeneration profile, t degen (θ). Equations (1–5) were solved at each iteration using the method of lines, with 51 or 101 mesh points. Insets show magnified portions of each graph. Numerical inverses are calculated and plotted only at those locations (eccentricities) where the analytical inverse fails to generate a t degen (θ) profile matching the target profile. Inverses resemble vertically flipped versions of the t degen (θ) profiles. Numerical solutions reveal lower values of the inverses near the fovea (θ = 0) than the analytical approximations suggest and higher values in some regions away from the fovea in (A–D) . Cone degeneration profile formulas and parameters are given in . Remaining parameter values as in .

Article Snippet: To find φ r inv (θ), we use the Matlab routine fminsearch (which uses a simplex search method), while to obtain f crit inv (θ) the Matlab routine patternsearch (which uses an adaptive mesh technique) was found to be more effective.

Techniques: Mutagenesis, Concentration Assay

Journal: Frontiers in Aging Neuroscience

Article Title: Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis

doi: 10.3389/fnagi.2022.765966

Figure Lengend Snippet: Inverse mutation-induced rod degeneration rate and TF threshold concentration—Pattern 3 target cone degeneration profiles. (A,C,E,G) inverse mutation-induced rod degeneration rate, ϕ r inv (θ) ( f crit = 3 × 1 0 - 5 ); (B,D,F,H) inverse TF threshold concentration, f crit inv (θ) ( ϕ r = 7.33 × 1 0 - 2 ). (A,B) linear 1 target cone degeneration profile, t degen (θ); (C,D) linear 2 t degen (θ) profile; (E,F) quadratic t degen (θ) profile; (G,H) cubic t degen (θ) profile. The solid black and dashed green curves correspond to Scaling 1 (α = 7.01 × 10 4 and β = 1.79 × 10 6 ), while the solid blue and dashed red curves correspond to Scaling 2 (α = 7.01 × 10 2 and β = 1.79 × 10 4 ). The black and blue solid curves are analytical approximations to the inverses, obtained by plotting Equations (7) and (10) respectively (A,C,E,G) , and Equations (8) and (11), respectively, (B,D,F,H) . The green and red dashed curves are numerical inverses, obtained by using the Matlab routines fminsearch (A,C,E,G) , and patternsearch (B,D,F,H) to calculate the ϕ r and f crit profiles for which the contour described by p c ( θ , t ) / p ~ c ( θ ) = 0.99 matches the target cone degeneration profile, t degen (θ). Equations (1–5) were solved at each iteration using the method of lines, with 26, 51 or 101 mesh points. Insets show magnified portions of each graph. Numerical inverses are calculated and plotted only at those locations (eccentricities) where the analytical inverse fails to generate a t degen (θ) profile matching the target profile. Inverses resemble vertically flipped versions of the t degen (θ) profiles. Numerical solutions reveal lower values of the inverses near the fovea (θ = 0) than the analytical approximations suggest and higher values in some regions away from the fovea in (C–F,H) . Cone degeneration profile formulas and parameters are given in . Remaining parameter values as in .

Article Snippet: To find φ r inv (θ), we use the Matlab routine fminsearch (which uses a simplex search method), while to obtain f crit inv (θ) the Matlab routine patternsearch (which uses an adaptive mesh technique) was found to be more effective.

Techniques: Mutagenesis, Concentration Assay