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Stock-Based Compensation (Details) - USD ($) |  | 3 Months Ended | 9 Months Ended |  |  |
--- | --- | --- | --- | --- | --- | ---
Jan. 01, 2024 | Jun. 30, 2024 | Jun. 30, 2023 | Jun. 30, 2024 | Jun. 30, 2023 | 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 |  | 0 | 339,000 | 1,225,669 | 469,512 |  |  |
Weighted-average grant date fair value (in Dollars per share) |  |  | $ 0.92 | $ 1.08 | $ 0.88 |  |  |
Total share based expenses (in Dollars) |  | $ 338,609 | $ 296,402 | $ 1,004,105 | $ 834,211 |  |  |
Total number of stock options outstanding |  | 2,814,096 |  | 2,814,096 |  |  | 1,708,427 |
Stock option vested |  | 127,583 | 69,947 | 232,494 | 282,172 |  |  |
Stock options, forfeited |  | 65,000 | 521 | 120,000 | 521 |  |  |
Options were exercised (in Dollars) |  |  |  |  |  |  |  |
Future issuance |  | 289,215 |  | 289,215 |  |  |  |
Unrecognized stock-based compensation (in Dollars) |  | $ 2,713,998 |  | $ 2,713,998 |  |  |  |
Weighted average period |  |  |  | 2 years 6 months |  |  |  |
Stock Options [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Total share based expenses (in Dollars) |  | 202,338 | 158,528 | $ 603,957 | 482,276 |  |  |
Restricted Stock Units (RSUs) [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Total share based expenses (in Dollars) |  | $ 136,271 | $ 137,874 | $ 400,148 | $ 351,935 |  |  |
Granted shares |  | 0 | 249,000 | 1,006,725 | 310,728 |  |  |
Weighted-average grant date fair value (in Dollars per share) |  |  | $ 1.59 | $ 1.03 | $ 1.6 |  |  |
Restricted stock units vest |  | 162,309 | 52,299 | 232,523 | 294,109 |  |  |
Restricted stock units forfeited |  |  |  |  |  |  |  |
Vesting Period [Member] | Restricted Stock Units (RSUs) [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Other vesting, percentage |  |  |  |  | 50.00% |  |  |
Minimum [Member] | Restricted Stock Units (RSUs) [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Vest over period |  |  |  |  | 1 year |  |  |
Maximum [Member] | Restricted Stock Units (RSUs) [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Vest over period |  |  |  |  | 3 years |  |  |
NeuroOne Medical Technologies Corporation [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Future issuance shares |  |  |  |  |  |  |  | 420,350
Inducement Plan [Member] | Minimum [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Equity incentive awards under inducement plan |  |  |  |  |  | 150,000 |  |
Inducement Plan [Member] | Maximum [Member] |  |  |  |  |  |  |  |
Stock-Based Compensation [Line Items] |  |  |  |  |  |  |  |
Equity incentive awards under inducement plan |  |  |  |  |  | 570,350 |  |

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