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Stock-Based Compensation (Details) - USD ($) |  | 3 Months Ended |  |  |
--- | --- | --- | --- | --- | ---
Jan. 01, 2024 | Dec. 31, 2023 | Dec. 31, 2022 | Nov. 09, 2023 | Sep. 30, 2023 | Oct. 04, 2021
Stock-Based Compensation [Line Items] |  |  |  |  |  |
Shares outstanding percentage |  | 13.00% |  |  |  |
Option plan shares | 1,051,556 |  |  |  |  |
Granted shares |  | 1,160,669 | 73,731 |  |  |
Weighted-average grant date fair value (in Dollars per share) |  | $ 1.08 | $ 0.66 |  |  |
Total share based expenses (in Dollars) |  | $ 308,638 | $ 300,181 |  |  |
Total number of stock options outstanding |  | 2,814,096 |  |  | 1,708,427 |
Stock option vested |  | 56,616 | 127,446 |  |  |
Stock options, forfeited |  | 55,000 | 0 |  |  |
Future issuance shares |  | 182,314 |  |  |  |
Unrecognized share based compensation (in Dollars) |  | $ 2,372,540 |  |  |  |
Weighted average period |  | 2 years 6 months |  |  |  |
Minimum [Member] | Inducement Plan [Member] |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |
Increase to inducement plan reserve for future stock issuances |  |  |  | 150,000 |  |
Maximum [Member] | Inducement Plan [Member] |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |
Increase to inducement plan reserve for future stock issuances |  |  |  | 570,350 |  |
NeuroOne Medical Technologies Corporation [Member] |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |
Future issuance shares |  |  |  |  |  | 420,350
Stock Options [Member] |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |
Total share based expenses (in Dollars) |  | $ 187,431 | $ 181,744 |  |  |
Restricted Stock Units (RSUs) [Member] |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |
Total share based expenses (in Dollars) |  | $ 121,207 | $ 118,437 |  |  |
RSU vested |  | 37,679 | 21,930 |  |  |

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