<PAGE>   1
                                                                    EXHIBIT 10.4

                                                                         CLASS I



                               SEVERANCE AGREEMENT


                  THIS AGREEMENT (the "Agreement"), made and entered into
effective as of May 1, 2001 (the "Effective Date"), is by and between
CYBERONICS, INC., a Delaware corporation (the "Company"), and SHAWN P. LUNNEY
(the "Employee").

                  WHEREAS, Employee is a key employee of the Company; and

                  WHEREAS, the Company recognizes that the possibility of a
Change of Control (as defined below) of the Company is unsettling and may result
in the departure of key employees to the detriment of the Company and its
stockholders; and

                  WHEREAS, the Board of Directors of the Company (the "Board")
has authorized this Agreement and certain similar agreements in order to retain
key employees to ensure the continuity of its management;

                  THEREFORE, for good and valuable consideration, the receipt
and sufficiency of which are hereby acknowledged, the Company and Employee agree
as follows:

         1. TERM. This Agreement shall commence on the Effective Date and shall
continue until April 30, 2003; provided, however, that commencing on April 30,
2002 and on each April 30th thereafter, the Term of this Agreement shall
automatically be extended for one additional year, unless at least six months
prior to such April 30 date the Board shall give written notice to Employee that
the Term of this Agreement shall cease to be so extended; provided further,
however, that if a Change of Control shall occur during the Term, the Term shall
automatically continue in effect for a period of not less than one year from the
date of such Change of Control. Notwithstanding the foregoing, except as
provided in Section 3, this Agreement shall automatically terminate on
Employee's termination of employment; provided, however, termination of this
Agreement shall not alter or impair any rights of Employee arising hereunder on
or prior to such termination.

         2. CHANGE OF CONTROL. For purposes of this Agreement, a Change of
Control of the Company shall mean:

                  (i) the acquisition by any "person," as such term is used in
Sections 13(d) and 14(d) of the Securities Exchange Act of 1934, as amended (the
"Exchange Act"), other than the Company, a subsidiary of the Company or a
Company employee benefit plan, of "beneficial ownership" (as defined in Rule
13d-3 under the Exchange Act), directly or indirectly, of securities of the
Company representing 50% or more of the combined voting power of the Company's
then outstanding securities entitled to vote generally in the election of
directors; or

<PAGE>   2

                  (ii) the consummation of a reorganization, merger,
consolidation or other form of corporate transaction or series of transactions,
in each case, with respect to which persons who were the shareholders of the
Company immediately prior to such reorganization, merger or consolidation or
other transaction do not, immediately thereafter, own more than 50% of the
combined voting power entitled to vote generally in the election of directors of
the reorganized, merged or consolidated company's then outstanding voting
securities in substantially the same proportions as their ownership immediately
prior to such event; or

                  (iii) the sale or disposition by the Company of all or
substantially all the Company's assets; or

                  (iv) a change in the composition of the Board, as a result of
which fewer than a majority of the directors are Incumbent Directors. "Incumbent
Directors" shall mean directors who either (A) are directors of the Company as
of October 2, 2000, or (B) are elected, or nominated for election, thereafter to
the Board with the affirmative votes of at least a majority of the Incumbent
Directors at the time of such election or nomination, but "Incumbent Director"
shall not include an individual whose election or nomination is in connection
with (i) an actual or threatened election contest (as such terms are used in
Rule 14a-11 of Regulation 14A promulgated under the Exchange Act) or an actual
or threatened solicitation of proxies or consents by or on behalf of a person
other than the Board or (ii) a plan or agreement to replace a majority of the
then Incumbent Directors; or

                  (v) the approval by the Board or the stockholders of the
Company of a complete or substantially complete liquidation or dissolution of
the Company.

         3. TERMINATION ON OR FOLLOWING A CHANGE OF CONTROL. If a Change of
Control occurs during the Term, Employee shall be entitled to the benefits
provided in Section 4 hereof if, during the Protected Period (as hereinafter
defined), Employee becomes disabled or Employee's employment is terminated,
unless such termination is (a) due to Employee's death, (b) by the Company
either for Cause or Employee's Disability, or (c) by Employee for other than a
Good Reason. Anything in this Agreement to the contrary notwithstanding, if
Employee's employment with the Company is terminated during the Term and prior
to the date on which a Change of Control occurs, and it is reasonably
demonstrated that such termination (i) was at the request of a third party who
has taken steps reasonably calculated to effect the Change of Control, or (ii)
otherwise arose in connection with or anticipation of the Change of Control,
then for all purposes of this Agreement the Change of Control shall be deemed to
have occurred on the date immediately prior to the date of Employee's
termination and Employee shall be deemed terminated by the Company during the
Protected Period other than for Cause. For purposes of this Agreement, the
"Protected Period" shall mean the period of time beginning with the Change of
Control and ending on the first anniversary of such Change of Control or
Employee's death, if earlier.

         (i) DISABILITY. If, as a result of Employee's incapacity due to
         physical or mental illness, Employee shall have been absent from
         Employee's duties with the Company on a full-time basis for 150
         consecutive calendar days, and within 30 days after written Notice of
         Termination (as defined hereinafter) Employee shall not have returned
         to the full-time performance of Employee's duties, the Company may
         terminate Employee's employment for "Disability"; provided, however, a
         termination of Employee's employment for Disability


                                       2
<PAGE>   3

         under this Agreement shall not alter or impair Employee's rights as a
         "disabled employee" under any of the Company's employee benefit plans.

         (ii) CAUSE. The Company may terminate Employee's employment for Cause.
         For the purposes of this Agreement, the Company shall have "Cause" to
         terminate Employee's employment hereunder only upon (A) the willful and
         continued failure by Employee to perform substantially Employee's
         duties with the Company, other than any such failure resulting from
         Employee's incapacity due to physical or mental illness, which
         continues unabated after a written demand for substantial performance
         is delivered to Employee by the Board that specifically identifies the
         manner in which the Board believes that Employee has not substantially
         performed Employee's duties or (B) Employee willfully engaging in gross
         misconduct that is materially and demonstrably injurious to the
         Company. For purposes of this paragraph, an act or failure to act on
         Employee's part shall be considered "willful" only if done or omitted
         to be done by Employee otherwise than in good faith and without
         reasonable belief that Employee's action or omission was in the best
         interest of the Company. Notwithstanding the foregoing, Employee shall
         not be deemed to have been terminated for Cause unless and until there
         shall have been delivered to Employee a copy of a resolution duly
         adopted by the affirmative vote of not less than three-quarters of the
         entire membership of the Board, at a meeting of the Board called and
         held for such purpose (after reasonable notice to Employee and an
         opportunity for Employee, together with Employee's counsel, to be heard
         before the Board), finding that in the good faith opinion of the Board
         Employee was guilty of conduct set forth in clauses (A) or (B) of this
         subsection (ii) and specifying the particulars thereof in reasonable
         detail.

         (iii) GOOD REASON. Employee may terminate Employee's employment for
         Good Reason. For purposes of this Agreement, "Good Reason" shall mean
         the occurrence of any of the following without Employee's express
         written consent:

                  (A) an adverse change (as determined by Employee in good
                  faith, which determination shall be controlling for all
                  purposes under this Agreement) in Employee's (i) positions,
                  duties, responsibilities or status with the Company from that
                  in effect immediately prior to the Change of Control, or (ii)
                  reporting responsibilities, titles or offices as in effect
                  immediately prior to the Change of Control; or any removal of
                  Employee from, or any failure to re-elect or appoint Employee
                  to, any of such responsibilities, titles, offices or
                  positions, except in connection with the termination of
                  Employee's employment for Cause or Disability, or as a result
                  of Employee's death, or by Employee for other than a Good
                  Reason;

                  (B) a reduction in Employee's annual rate of base salary as in
                  effect immediately prior to the Change of Control or as the
                  same may be increased from time to time thereafter (the "Base
                  Salary");

                  (C) a failure by the Company to continue the Company's Annual
                  Incentive Compensation Plan as the same may be modified from
                  time to time, but substantially in the form in effect
                  immediately prior to the Change of Control (the "Bonus Plan"),
                  or a failure by the Company to continue Employee as a
                  participant in the Bonus Plan


                                       3
<PAGE>   4
                  in at least the same amount (the "Bonus Amount") as Employee's
                  target bonus amount under the Bonus Plan with respect to the
                  fiscal year ending immediately prior to the Change of Control
                  or with respect to the current fiscal year if Employee has
                  been employed by the Company for a shorter period (Bonus
                  Amounts related to less than a full fiscal year shall be
                  annualized for this purpose);

                  (D) the failure by the Company to continue in effect any other
                  employee benefit or compensation plan program or policy, in
                  which Employee is participating immediately prior to the
                  Change of Control, unless the Company establishes such new
                  plans, programs or policies as is necessary to provide
                  Employee with substantially comparable benefits; the taking of
                  any action by the Company not required by law that would
                  adversely affect Employee's participation in or reduce
                  Employee's benefits under any of such plans, programs or
                  policies or deprive Employee of any material fringe benefit
                  enjoyed by Employee immediately prior to the Change of
                  Control;

                  (E) the Company's requiring Employee to relocate to an office
                  more than 25 miles from the Company's office to which Employee
                  was assigned immediately prior to the Change of Control,
                  except for required travel on the Company's business to an
                  extent substantially consistent with Employee's business
                  travel obligations immediately prior to the Change of Control;

                  (F) the amendment, modification or repeal of any provision of
                  the Company's Certificate of Incorporation, as amended, or the
                  Bylaws of the Company which was in effect immediately prior to
                  such Change of Control, if such amendment, modification or
                  repeal would adversely effect Employee's right to
                  indemnification by the Company;

                  (G) the failure of the Company to obtain the assumption of
                  this Agreement by any successor as contemplated in Section 6
                  hereof; or

                  (H) any purported termination of Employee's employment that is
                  not effected pursuant to a Notice of Termination satisfying
                  the requirements of subparagraph (iv) below and, if
                  applicable, subparagraph (ii) above; and for purposes of this
                  Agreement, no such purported termination shall be effective.

         Employee's right to terminate employment for a Good Reason hereunder
         shall not be affected by Employee's incapacity due to a physical or
         mental illness nor shall Employee's continued employment following any
         circumstance that constitutes a Good Reason hereunder, regardless of
         the length of such continued employment, constitute a consent to or a
         waiver of Employee's rights hereunder with respect to such
         circumstance.

         (iv) NOTICE OF TERMINATION. Any termination by the Company pursuant to
         subparagraphs (i) or (ii) above, or by Employee pursuant to
         subparagraph (iii) above, shall be communicated by written Notice of
         Termination to the other party hereto. For purposes of this Agreement,
         a "Notice of Termination" shall mean a notice that shall indicate the


                                       4
<PAGE>   5

         specific termination provision in this Agreement relied upon and shall
         set forth in reasonable detail the facts and circumstances claimed to
         provide a basis for termination of Employee's employment under the
         provision so indicated.

         (v) DATE OF TERMINATION. "Date of Termination" shall mean (A) if
         Employee is terminated for Disability, 30 days after Notice of
         Termination is given, provided that Employee shall not have returned to
         the performance of Employee's duties on a full-time basis during such
         30-day period, (B) if Employee's employment is terminated pursuant to
         subparagraph (iii) above, the date specified in the Notice of
         Termination, (C) with respect to a termination prior to a Change of
         Control, which is deemed to be after such Change of Control as provided
         in Section 3, the date of such termination, and (D) if Employee's
         employment is terminated for any other reason on or after a Change of
         Control, the date of such termination.

         4.       COMPENSATION DURING DISABILITY OR UPON TERMINATION.

         (i) If, during the Protected Period, Employee fails to perform
         Employee's normal duties as a result of incapacity due to physical or
         mental illness, Employee shall continue during the period of such
         disability to receive Employee's full Base Salary and any awards,
         deferred and nondeferred, payable during such period under the Bonus
         Plan, less any amounts paid to Employee during such period of
         disability pursuant to the Company's short term disability or
         sick-leave program(s) until Employee's employment is terminated or such
         Disability ends. This Section 4(i) shall not reduce or impair
         Employee's rights to terminate employment for a Good Reason as
         otherwise provided herein.

         (ii) If, during the Protected Period, Employee's employment shall be
         terminated (x) by the Company for Cause, (y) by Employee's death, or
         (z) by Employee other than for a Good Reason, the Company shall pay
         Employee's earned but unpaid Base Salary through the Date of
         Termination and the Company shall have no further obligations to
         Employee under this Agreement.

         (iii) If, during the Protected Period, (1) the Company shall terminate
         Employee other than for Cause or Disability or (2) Employee shall
         terminate Employee's employment for a Good Reason, the Company shall
         pay to Employee, by certified or bank cashier's check or wire transfer
         within five business days after the Date of Termination, an amount
         equal to: (A) three times the sum of Employee's Base Salary and Bonus
         Amount; plus (B) that portion of Employee's Base Salary earned, and
         vacation pay vested for the prior year and accrued for the current year
         to the Date of Termination, but not paid or used, and all other amounts
         previously deferred by Employee or earned but not paid as of such date
         under all Company bonus or pay plans or programs.

         (iv) If any payment due under the terms of this Agreement is not timely
         made or otherwise withheld by the Company, its successors or assigns,
         interest shall accrue on such payment at the highest maximum legal rate
         permissible under applicable law from the date such payment first
         became due through the date of payment thereof.



                                       5
<PAGE>   6

         (v) In the event that any payment or benefit received or to be received
         by Employee pursuant to the terms of this Agreement or any other plan,
         arrangement or agreement with (A) the Company, (B) any Person whose
         actions result in a "change in control" (for purposes of Section 280G
         of the Internal Revenue Code (the "Code")) or (C) any Person affiliated
         with the Company or such Person) (all such payments and benefits being
         hereinafter called "Total Payments") would be subject to the excise tax
         imposed under Section 280G of the Code, the Company shall pay to
         Employee such additional amount (the "Gross-Up Payment") such that the
         net amount retained by Employee, after deduction of any excise tax
         imposed under Section 4999 of the Code (the "Excise Tax") on the Total
         Payments and all federal, state and local taxes, including the Excise
         Tax, upon the Gross-Up Payment, shall be equal to the Total Payments.
         For purposes of determining the amount of the Gross-Up Payment,
         Employee shall be deemed to pay federal income tax at the highest
         marginal rate of federal income taxation in the calendar year in which
         the Gross-Up Payment is to be made and state and local income taxes at
         the highest marginal rate of taxation in the state and locality of
         Employee's residence on the date on which the Gross-Up Payment is
         calculated for purposes of this subparagraph. In the event that the
         Excise Tax is subsequently determined to be less than the amount taken
         into account hereunder, Employee shall repay to the Company, at the
         time that the amount of such reduction in Excise Tax is finally
         determined, the portion of the Gross-Up Payment attributable to such
         reduction (plus that portion of the Gross-Up Payment being repaid by
         Employee to the extent that such repayment results in a reduction in
         Excise Tax and/or a federal, state or local income tax deduction) plus
         interest on the amount of such repayment at the rate provided in
         Section 1274(b)(2)(B) of the Code. In the event that the Excise Tax is
         determined to exceed the amount taken into account hereunder (including
         by reason of any payment the existence or amount of which cannot be
         determined at the time of the Gross-Up Payment), the Company shall make
         an additional Gross-Up Payment in respect of such excess (plus any
         interest, penalties or additions payable by Employee with respect to
         such excess) at the time that the amount of such excess if finally
         determined. Employee and the Company shall each reasonably cooperate
         with the other in connection with any administrative or judicial
         proceedings concerning the existence or amount of liability for Excise
         Tax with respect to the Total Payments. The parties intend that the
         Gross-Up Payment be determined in a manner that is most favorable to
         Employee.

         5. NO MITIGATION OR OFFSET. The provisions of this Agreement are not
intended to, nor shall they be construed to, require that Employee mitigate the
amount of any payment provided for in this Agreement by seeking or accepting
other employment, nor shall the amount of any payment provided for in this
Agreement be reduced by any compensation earned by Employee as the result of
employment by another employer or otherwise. Without limitation of the
foregoing, the Company's obligations to make the payments to Employee required
under this Agreement shall not be affected by any set off, counterclaim,
recoupment, defense or other claim, right or action that the Company may have
against Employee.

         6.       SUCCESSORS; BINDING AGREEMENT.

         (i) The Company will require any successor, whether direct or indirect,
         by purchase, merger, consolidation or otherwise, to all or
         substantially all of the business and/or assets of



                                       6
<PAGE>   7

         the Company, by agreement in form and substance reasonably satisfactory
         to Employee, expressly to assume and agree to perform this Agreement in
         the same manner and to the same extent as the Company would have been
         required if no such succession had taken place. Failure of the Company
         to obtain such agreement prior to the effectiveness of any such
         succession shall be a breach of this Agreement and shall entitle
         Employee to payment from the Company in the same amount and on the same
         terms as Employee would be entitled hereunder if Employee had
         terminated Employee's employment for Good Reason, except that for
         purposes of implementing the foregoing, the date on which any such
         succession becomes effective shall be deemed the Date of Termination.
         As used in this Agreement, "Company" shall mean the Company as
         hereinbefore defined and any successor to its business and/or assets as
         aforesaid that executes and delivers the agreement provided for in this
         Section 6 or which otherwise becomes bound by all the terms and
         provisions of this Agreement by operation of law.

         (ii) This Agreement shall inure to the benefit of and be enforceable by
         Employee's personal or legal representatives, executors,
         administrators, successors, heirs, distributees, devisees and legatees.
         If Employee should die while any amounts would still be payable to
         Employee hereunder if Employee had continued to live, all such amounts
         shall be paid in accordance with the terms of this Agreement to
         Employee's beneficiary as filed with the Company pursuant to this
         Agreement or, if there be no such designated beneficiary, to Employee's
         estate.

         7. NOTICE. All notices, consents, waivers, and other communications
required under this Agreement must be in writing and will be deemed to have been
duly given when (a) delivered by hand (with written confirmation of receipt),
(b) sent by facsimile (with confirmation of receipt), provided that a copy is
mailed by certified mail, return receipt requested, or (c) when received by the
addressee, if sent by a nationally recognized overnight delivery service, in
each case to the appropriate addresses and facsimile numbers set forth below (or
to such other addresses and facsimile numbers as a party may designate by notice
to the other parties):


         If to the Company:

                  Cyberonics, Inc.

                  ----------------------------
                  ----------------------------
                  Facsimile No.:
                                 -------------

         If to Employee:

                  ----------------------------
                  ----------------------------
                  ----------------------------


                                       7
<PAGE>   8

         8. MISCELLANEOUS. No provisions of this Agreement may be modified,
waived or discharged unless such waiver, modification or discharge is agreed to
in writing signed by Employee and by the Chairman of the Board or an authorized
officer of the Company. No waiver by either party hereto at any time of any
breach by the other party hereto of, or compliance with, any condition or
provision of this Agreement to be performed by such other party shall be deemed
a waiver of similar or dissimilar provisions or conditions at the same or at any
prior or subsequent time.

         9. VALIDITY. The interpretation, construction and performance of this
Agreement shall be governed by and construed and enforced in accordance with the
laws of the State of Texas without regard to conflicts of laws principles. The
invalidity or unenforceability of any provisions of this Agreement shall not
affect the validity or enforceability of any other provision of this Agreement,
each of which shall remain in full force and effect.

         10. COUNTERPARTS. This Agreement may be executed in one or more
counterparts, each of which shall be deemed to be an original but all of which
together shall constitute one and the same instrument.

         11. DESCRIPTIVE HEADINGS. Descriptive headings are for convenience only
and shall not control or affect the meaning or construction of any provision of
this Agreement.

         12. CORPORATE APPROVAL. This Agreement has been approved by the Board,
and has been duly executed and delivered by Employee and on behalf of the
Company by its duly authorized representative.

         13. DISPUTES. The parties agree to resolve any claim or controversy
arising out of or relating to this Agreement by binding arbitration under the
Federal Arbitration Act before one arbitrator in the City of Houston, State of
Texas, administered by the American Arbitration Association under its Commercial
Arbitration Rules, and judgment on the award rendered by the arbitrator may be
entered in any court having jurisdiction thereof. The Company shall reimburse
Employee, on a current basis, for all legal fees and expenses incurred by
Employee in connection with any dispute arising under this Agreement, including,
without limitation, the fees and expenses of the arbitrator, unless the
arbitrator finds Employee brought such claim in bad faith, in which event each
party shall pay its own costs and expenses and Employee shall repay to the
Company any fees and expenses previously paid on Employee's behalf by the
Company.

                  The parties stipulate that the provisions hereof shall be a
complete defense to any suit, action, or proceeding instituted in any federal,
state, or local court or before any administrative tribunal with respect to any
controversy or dispute arising during the period of this Agreement and which is
arbitrable as herein set forth. The arbitration provisions hereof shall, with
respect to such controversy or dispute, survive the termination of this
Agreement.


                                       8
<PAGE>   9

                  IN WITNESS WHEREOF, the Company and Employee have executed
this Agreement in multiple counterparts effective for all purposes as of the
Effective Date.


                                        CYBERONICS, INC.

                                        By: s/s Robert P. Cummins
                                        Name: Robert P. Cummins
                                        Title: President & CEO


                                        EMPLOYEE


                                        s/s Shawn P. Lunney


                                       9