CAPNIA, INC. AMENDMENTNO. 2 TO CONVERTIBLE PROMISSORY NOTES AND WARRANTS TO PURCHASE SHARES

Exhibit 10.13

CAPNIA, INC.

AMENDMENT NO. 2 TO CONVERTIBLE PROMISSORY NOTES AND

WARRANTS TO PURCHASE SHARES

This Amendment No. 2 to Convertible Note and Warrant Purchase Agreement, Convertible Promissory Notes and Warrants to Purchase Shares (this “Amendment”) is made and entered into as of January 17, 2012 (the “Effective Date”), by and among Capnia, Inc., a Delaware corporation (the “Company”), with offices at 2445 Faber Place, Suite 250, Palo Alto, CA 94303 and the persons and entities who are signatories hereto (the “Investors”).

RECITALS

A. WHEREAS, the Company and the Investors are parties to that certain Convertible Promissory Note and Warrant Purchase Agreement, dated as of February 10, 2010, as amended by the Amendment No. 1 to Convertible Promissory Note and Warrant Purchase Agreement, Convertible Promissory Notes and Warrants to Purchase Shares, dated as of November 10, 2010 (as amended, the “2010 Purchase Agreement”), pursuant to which the Company sold and issued to each Investor, a convertible promissory note in the principal amount set forth opposite such Investor’s name on Exhibit A thereto (together with all such convertible promissory notes sold and issued pursuant to the 2010 Purchase Agreement, the “2010 Notes”), together with a corresponding warrant to purchase shares of the Company’s capital stock (together with all such warrants to purchase shares issued pursuant to the Agreement, the “2010 Warrants”);

B. WHEREAS, Section 6.1 of the 2010 Purchase Agreement provides that the holders of at least two-thirds (2/3) in interest of the 2010 Notes may, with the Company’s prior written consent, waive, modify or amend any provisions of the 2010 Purchase Agreement, the 2010 Notes and the 2010 Warrants on behalf of all Investors (as defined in the 2010 Purchase Agreement).

C. WHEREAS, pursuant to the Note and Warrant Purchase Agreement, dated as of even date herewith, by and among the Company and the persons and entities listed on the Schedule of Investors attached thereto as Exhibit A (the “2012 Purchase Agreement”), the Company is undertaking the offering and sale of up to $2,000,000 in principal amount of new convertible promissory notes (the “2012 Notes”), together with related warrants to purchase shares of the Company’s capital stock that will be issued pursuant to the 2012 Purchase Agreement.

D. WHEREAS, the Company and the undersigned Investors, representing the holders of at least two-thirds (2/3) in interest of the 2010 Notes, now desire to amend the 2010 Notes and the 2010 Warrants to: (i) clarify and confirm that the 2010 Notes, issued pursuant to the 2010 Purchase Agreement, shall rank pari passu with the 2012 Notes, issued pursuant to the 2012 Purchase Agreement, in all respects; (ii) extend the Maturity Date of the 2010 Notes to January 17, 2013, such that the 2010 Notes and the 2012 Notes shall have the same maturity date; (iii) amend the definition of Shares under the 2010 Warrants to reflect the extended Maturity Date of the 2010 Notes; and (iv) make certain other changes.

E. WHEREAS, unless otherwise stated, capitalized terms not otherwise defined herein shall have the respective meanings ascribed to such terms in the 2010 Purchase Agreement, 2010 Notes and 2010 Warrants, as applicable.

AGREEMENT

NOW THEREFORE, in consideration of the foregoing, and the representations, warranties, and conditions set forth below, the parties hereto, intending to be legally bound, hereby agree as follows:

1. Amendments.

1.1 Amendments to 2010 Notes.

(a) The first paragraph of each 2010 Note is hereby amended and restated in its entirety to read as follows:

“FOR VALUE RECEIVED CAPNIA, INC., a Delaware corporation (the “Company”), promises to pay to                     (the “Holder”), or its registered assigns, the principal amount of $            , or such lesser amount as shall equal the outstanding principal amount hereof, together with compound interest from the date of this Note on the unpaid principal balance at a rate equal to 12% per annum, compounded on the first day of each month and computed on the basis of the actual number of days elapsed and a year of 365 days. Two (2) times the unpaid principal, together with any then accrued but unpaid interest and any other amounts payable hereunder, shall be due and payable on the earlier of: (i) upon demand made after January 17, 2013 (the “Maturity Date”) by Holders representing at least a majority of the principal amount of all then outstanding Notes issued pursuant to that certain Convertible Note and Warrant Purchase Agreement by and among the Company and the Investors described therein, dated as of February 10, 2010, (as the same may from time to time be amended, modified or supplemented, the “Purchase Agreement”); or (ii) when, upon or after the occurrence of an Event of Default (as defined below), such amounts are declared due and payable by the Holder or made automatically due and payable in accordance with the terms of this Note. This Note is one of the “Notes” issued pursuant to the Purchase Agreement. Capitalized terms not otherwise defined herein shall have the meanings set forth in the Purchase Agreement.”

(b) The following is inserted as the new Section 5(j) of each 2010 Note:

“(j) Pari Passu Notes. The Holder acknowledges and agrees that the payment of all or any portion of the outstanding principal amount of this Note, and all accrued and unpaid interest under this Note, shall be pari passu in right of payment and in all other respects to: (i) the other Notes; and (ii) the convertible promissory notes (the “2012 Notes”) issued pursuant to the Convertible Note and Warrant Purchase Agreement, dated as of January 17, 2012, by and among the Company and the persons and entities listed on the Schedule of Investors attached thereto as Exhibit A. In the event that the Holder receives payments in excess of such Holder’s pro rata share of the Company’s payments to the holders of all the Notes and the 2012 Notes, then the

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Holder shall hold in trust all such excess payments for the benefit of the holders of all other Notes and the 2012 Notes, and shall pay such amounts held in trust to such other holders upon demand by such holders.”

1.2 Amendment to Warrant to Purchase Shares. Section 1(a) of each Warrant is hereby amended and restated in its entirety to read as follows:

“(a) Definition of Shares. “Shares” shall mean Next Financing Securities in the event that a Next Financing occurs prior to January 17, 2013 (the “Note Maturity Date”) and the Holder converts the Note issued to the Holder into Next Financing Securities. In the event that a Next Financing has not occurred before the Note Maturity Date, “Shares” shall mean either the Company’s Series C Preferred Stock or Next Financing Securities, whichever such securities’ Original Issue Price (as such term is defined in the Company’s Amended and Restated Certificate of Incorporation) is lower.”

2. Miscellaneous.

2.1 Waivers and Amendments. This Amendment nor any term hereof may be amended, waived, discharged or terminated other than by a written instrument signed by the party against whom enforcement of any such amendment, waiver, discharge or termination is sought; provided, however, that the holders of at least two-thirds in interest of the Notes issued pursuant to the Agreement may, with the Company’s prior written consent, waive, modify or amend any provisions hereof on behalf of all Investors.

2.2 Governing Law. This Amendment shall be governed in all respects by the laws of the State of California as such laws are applied to agreements between California residents entered into and to be performed entirely within California.

2.3 Successors and Assigns. Except as otherwise expressly provided herein, the provisions hereof shall inure to the benefit of, and be binding upon, the successors, assigns, heirs, executors and administrators of the parties hereto.

2.4 Entire Agreement. This Amendment (including the exhibits attached hereto and the other documents delivered pursuant hereto) and, to the extent not amended hereby, the 2010 Purchase Agreement, the 2010 Notes, the 2010 Warrants and the exhibits attached thereto and the other documents delivered pursuant thereto, constitute the full and entire understanding and agreement between the parties with regard to the subjects hereof and thereof.

2.5 Legal Effect. This Amendment shall constitute an integral part of the the 2010 Notes and the 2010 Warrants. Except as set forth in this Amendment, the 2010 Notes and the 2010 Warrants shall continue in full force and effect in accordance with their respective terms.

2.6 Counterparts. This Amendment may be executed in any number of counterparts, each of which shall be an original, but all of which together shall be deemed to constitute one instrument.

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2.7 Facsimile; Execution and Delivery. A facsimile or other reproduction of this Amendment may be executed by one or more parties hereto and delivered by such party by facsimile or any similar electronic transmission device pursuant to which the signature of or on behalf of such party can be seen. Such execution and delivery shall be considered valid, binding and effective for all purposes. At the request of any party hereto, all parties hereto agree to execute and deliver an original of this Amendment as well as any facsimile, telecopy or other reproduction hereof.

(Signature Pages Follow)

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IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]

IN WITNESS WHEREOF, the parties have caused this Amendment No. 2 to Convertible Promissory Notes and Warrants to Purchase Shares to be duly executed and delivered by their proper and duly authorized officers as of the date and year first written above.

[Signature Page to Amendment No. 2 to Convertible Promissory Notes

and Warrants to Purchase Shares]