EXHIBIT 32.1

CERTIFICATION OF THE CHIEF EXECUTIVE OFFICER AND

CHIEF FINANCIAL OFFICER OF CYBERONICS, INC.

PURSUANT TO 18 U.S.C. SECTION 1350

      Robert P. Cummins, the Chief Executive Officer of Cyberonics, Inc. (the “Company”), and Pamela B. Westbrook, the Chief Financial Officer of Cyberonics, Inc. each hereby certifies that:

      (a) the accompanying report on Form 10-K for the fiscal year ended April 30, 2004 as filed with the Securities and Exchange Commission on the date hereof (the “Report”) by the Company fully complies with the requirements of section 13(a) or 15(d) of the Securities Exchange Act of 1934, as amended; and

      (b) the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

Date: July 9, 2004

/s/ ROBERT P. CUMMINS

Robert P. Cummins

Chairman of the Board, President and

Chief Executive Officer

(Principal Executive Officer)

/s/ PAMELA B. WESTBROOK

Pamela B. Westbrook

Vice President of Finance and Administration and

Chief Financial Officer

(Principal Financial and Accounting Officer)

      The certification set forth above is being submitted to the Securities and Exchange Commission solely for the purpose of complying with Section 1350 of Chapter 63 of Title 18 of the United States Code. This certification is not to be deemed filed for purposes of Section 18 of the Securities Exchange Act of 1934, as amended, and does not constitute a part of the Report accompanying this letter.

      A signed original of this written statement required by Section 906 has been provided to Cyberonics, Inc. and will be retained by Cyberonics, Inc. and furnished to the Securities and Exchange Commission or its staff upon request.